What simple steps do I take to create a financial plan?
How do I use financial planning in everyday life?
How is the financial planning process implemented for every purchase?
If you fail to plan, you are planning to fail.
Honestly, practicing money management isn’t that hard to figure out. In many ways it’s similar to playing a video game. The first time you play a game, you may feel awkward or have the lowest score. Playing for a while can make you OK at the game. But if you learn the rules of the game, figure out how to best use each tool in the game, read strategy guides from experts, and practice, you can get really good at it.
Money management is the same. It’s not enough to “figure it out as you go.” If you want to get good at managing your money, you must treat money like you treat your favorite game. You have to come at it with a well-researched plan.

The amount of debt I have or will have
Getting a job that will pay well enough
Being financially independent
Supporting my family
Planning/saving for the future
When considering how to pay for college, which of the following do you know
least
about?
Grants
Scholarships
Loans
Work-study programs
You can also take the anonymous What Students Say surveys to add your voice to this textbook. Your responses will be included in updates.
You can also take the anonymous
What Students Say surveys
to add your voice to this textbook. Your responses will be included in updates.
Students offered their views on these questions, and the results are displayed in the graphs below.
What is your top immediate financial priority?
Which aspect of your finances concerns you the most?
When considering how to pay for college, which of the following do you know
least
about?
Financial Planning Process
Personal goals and behaviors have a financial component or consequence. To make the most of your financial resources, you need to do some financial planning. The financial planning process consists of five distinct steps: goal setting, evaluating, planning, implementing, and monitoring. You can read in more depth about SMART goals in chapter 3.
Financial Planning in Five Steps
Develop Personal Goals
What do I want my life to look like?
Identify and Evaluate Alternatives for Achieving Goals for My Situation
What do my savings, debt, income, and expenses look like?
What creative ways are available to get the life I want?
Write My Financial Plan
What small steps can I take to start working toward my goals?
Implement the Plan
Begin taking those steps, even if I can only do a few small things each week.
Monitor and Adjust the Plan
Make sure I don’t get distracted by life. Keep taking those small steps each week. Make adjustments when needed.
How to Use Financial Planning in Everyday Life
The financial planning process isn’t only about creating one big financial plan. You can also use it to get a better deal when you buy a car or computer or rent an apartment. In fact, anytime you are thinking about spending a lot of money, you can use the financial planning process to pay less and get more.
To explore financial planning in depth, we’ll use the example of buying a car.
1. Develop Goals
First, what do you really need? If you’re looking for a car, you probably need transportation.  Could you take a bus, walk, or bike instead? Often one goal can impact another goal. Cars are typically not good financial investments. We have cars for convenience and necessity, to earn an income and to enjoy life. Financially, they are an expense. They lose value, or depreciate, rather than increasing in value, like savings. So buying a car may slow your savings or retirement plan goals. Cars continually use up cash for gas, repairs, taxes, parking, and so on. Keep this in mind throughout the planning process.
2. Identify and Evaluate Alternatives for Achieving Goals in Your Current Situation.
For this example, let’s assume that you have determined the best alternative is to buy a car. Do you need a new car? Will your current car last with some upkeep? Consider a used car over a new one. On average, a new car will lose one-fifth of its value during its first year.
5
Buying a one-year-old car is like getting a practically new car for a 20 percent discount. So in many cases, the best deal may be to buy a five- or six-year-old car. Sites such as the Kelley Blue Book website (KBB.com) and Edmunds.com can show you depreciation tables for the cars you are considering. Perhaps someone in your family has a car they will sell you at a discount.
Do you know how much it will cost in total to own the car? It will help to check out the total cost of ownership tools (also on KBB.com and Edmunds.com) to estimate how much each car will cost you in maintenance, repairs, gas, and insurance. A cheap car that gets poor gas mileage and breaks down all the time will actually cost you more in the long run.
3. Write Down Your Financial Plan
Table 10.1 Examples of financial plans for a car and a computer.
Goal
Item
Details
Budget
Timeline
Transportation/Car
2014 Toyota Camry
Black, A/C, power windows, less than 60,000 miles
Car $12,000 (max)
Down payment $3,000
Insurance $100/mo
Sales tax $900
+ Licensing $145
Cash needed $4,145
Have $3600 in savings for this.
Save $50/week.
Purchase in approximately 11 weeks.
Computer
Used or refurbished laptop
Dell w/ Windows, minimum 13", 128G hard drive, HD Graphics
$300
Use free Windows update from school.
Use free Wi-Fi at school.
Sell current laptop for $100.
Buy refurbished from Dell site for $289.
$189 on credit card.
Pay off when statement comes.
4. Implement Your Plan
Once you’ve narrowed down which car you are looking for, do more online research with resources such as Kelley Blue Book to see what is for sale in your area. You can also begin contacting dealerships and asking them if they have the car you are looking for with the features you want. Ask the dealerships with the car you want to give you their best offer, then compare their price to your researched price. You may have to spend more time looking at other dealerships to compare offers, but one goal of online research is to save time and avoid driving from place to place if possible.
When you do go to buy the car, bring a copy of your written plan into the dealership and stick to it. If a dealership tries to switch you to a more expensive option, just say no, or you can leave to go to another dealership. Remember Elan in our opening scenario? He went shopping alone and caved to the pressure and persuasion of the salesperson. If you feel it is helpful, take a responsible friend or family member with you for support.
5. Monitor and Adjust the Plan to Changing Circumstances and New Life Goals
Life changes, and things wear out. Keep up the recommended maintenance on the car (or any other purchase). Keep saving money for your emergency fund, then for your next car. The worst time to buy a car is when your current car breaks down, because you are easier to take advantage of when you are desperate. When your car starts giving you trouble or your life circumstances start to change, you will be ready to shop smart again.
A good practice is to keep making car payments once the car loan is paid off. If you are paying $300 per month for a car loan, when the loan is paid off, put $300 per month into a savings account for a new car instead. Do it long enough and you can buy your next car using your own money!
Use the Financial Planning Process for Everything
The same process can be used to make every major purchase in your life. When you rent an apartment, begin with the same assessment of your current financial situation, what you need in an apartment, and what goals it will impact or fulfill. Then look for an apartment using a written plan to avoid being sold on a more expensive place than you want.
You can even use the process of assessing and planning for small things such as buying textbooks or weekly groceries. While saving a few bucks each week may seem like a small deal, you will gain practice using the financial planning process, so it will become automatic for when you make the big decisions in life. Stick to your plan.


2.2: Using and Understanding Graphs
- Assess graphs for information and quality
Introduction to Graphs
Florence Nightingale \((1820-1910),\) although primarily known as a nurse, analyzed data in the service of her patients. She was the first woman to be a fellow of the Royal Statistical Society . She was one of the first nurses to use graphical representations to illustrate the causes of mortality. She created the graph below to share the true cause of British soldier mortality during the Crimean War clearly.
Figure \(\PageIndex{1}\): Polar-area diagram for British soldier mortality from April \(1854\) through March \(1855\)
From the graph, we know that causes of mortality are broken into three classifications: wounds in battle, disease, and other causes. We can conclude relatively quickly that being wounded in battle was not the leading cause of death; disease caused most of the deaths. When Nightingale arrived, the conditions of the military hospitals were awful, and more people died from disease than their wounds. Once her sanitation practices were implemented, the mortality rate dropped from \(42.7\%\) to \(2.2\%.\) The question we must address is: how do we trust that the graph relays the truth?
We were not provided any information about the construction of the graph; please be careful of graphs without enough explanation. We guessed that the number of mortalities was related to the slice size for any given month. This graph has two components to size: the radius and the area. By which measure are we making a comparison? Most of us naturally make mental comparisons based on the area of each slice. If the graph were constructed using the radius to indicate the number of mortalities, we could still tell which months had more deaths, but we would misjudge the magnitude of these differences. Nightingale constructed the graph so that the areas represented the amount of mortalities; this information is necessary for quality interpretation and understanding.
Graphs and charts are excellent ways to share information quickly with a larger audience. In this section, we will look at several different types of graphs. However, our goal is to identify certain types of graphs to become informed consumers of information and critical thinkers actively engaged in the world around us.
The pie graph below shows the percentage of visits to social media sites in \(2017.\) There are various issues with this particular pie graph; identify some of the problems.
- Answer
-
Pie graphs effectively display the relative frequencies of a small number of categories. However, pie graphs with a large number of categories are not recommended. This example uses too many categories.
Pie graphs are most commonly used to compare parts to the whole. This example has two components working against this comparison. The pie slices are not touching and the visual comparison of areas is skewed because of the three-dimensional formatting. When adding the third dimension to a graph, it must add information and not hide important information.
Furthermore, the numbers here do not add to \(100.\) This means that the number given does not represent the relative size of the slice of the pie. For example, Facebook appears to take up nearly half of the pie, but the number given is \(39.14.\) This makes it seem as if the websites listed here account for all the websites that see a relatively non-negligible amount of visits. There should be another slice, perhaps labeled "other," which takes up \(16.1\%\) of the pie.
Pie graphs are most helpful in communicating relative sizes or proportions of a few categories. They are not useful if there are numerous categories or if one wants to communicate absolute, not relative, measures. Note that relative measures are more likely to be misleading when the sample size is small.
A time series graph consists of the measurement of the same variable of the same subject taken over regular time intervals for a given period. Time series data frequently occurs in contexts where variables change: stock market prices, national economic figures, population tracking, etc. During the COVID-\(19\) pandemic, time series graphs were frequently used to communicate transmission and mortality rates. Such graphs can serve as natural and valuable visual aids, but they can be misleading if not constructed appropriately.
In August \(2020,\) Dr. Norman, the Secretary of the Kansas Department of Health and Environment, used the following graph to show the number of COVID-\(19\) cases in counties with mask mandates vs counties without mask mandates. Study the graph and see what conclusions can be made. Are there any issues?
- Answer
-
First, notice that the scales on each side of the graph represent the same information, but the numbers differ. The line representing \(25\) cases for masked counties (scale on the left) corresponds to \(14\) cases for unmasked counties (scale on the right). This \(11\) point difference is consistent throughout the scale. Without careful attention, a consumer might conclude that the \(7\)-day rolling averages for masked counties dropped below those of the unmasked counties. This, however, is not the case. Consider the following graph, where the same data is plotted using the same axes.
We now see clearly that the masked county \(7\)-day rolling averages stay above those of the unmasked counties. There can be a variety of factors at play in this difference. While the \(7\)-day rolling averages should be scaled to account for differences in population, other related factors could also be at play. Smaller and rural counties were less prone to mask mandates. Larger and more densely populated counties were more likely to have mask mandates. These differences alone could explain the differences in the \(7\)-day rolling averages.
The changes in the \(7\)-day rolling averages could be a better measure for the success of the mandates, but notice how the inclusion of \(0\) on the vertical scale helps us better gauge the change over time. The variation in the \(7\)-day rolling averages seems less dramatic in the second graph than in the original. The no-mask counties seem to hover around \(9\) or \(10\) while the mask counties seem to hover between \(16\) and\(19.\) The first data point makes the change seem quite stark, but a change in a \(7\)-day rolling average indicates that the first measurement was significantly higher than the subsequent days. Would one day of wearing masks have such an immediate effect? Possibly. Such a question warrants further analysis, preferably with each day's raw data.
We have encountered several bar graphs in the previous section as we studied relative frequency and frequency distributions. While pie graphs help make comparisons from a part to the whole, bar charts help make comparisons between parts and across different distributions.
The \(2009-2011\) data from the U.S. Census Survey of Income and Program Participants was released in USA Today in \(2012,\) and the bar graph below is used. At first glance, the information on the increased number of people on welfare over two years is staggering. What else do you notice about this graph?
- Answer
-
Although having \(108\) million people on welfare is not good, we must recognize in this particular graph that the vertical axis begins at \(94\) million, making the increase look more extreme than it is. Be sure to check the vertical axis to see where it begins. We naturally make mental length comparisons and ratios when looking at bar charts, but recall that ratios are only meaningful if we have a meaningful zero value. When bar charts do not start at a meaningful zero value, like in the graph above, the mental ratio comparisons have no meaning.
We remark here that bar charts displaying ratio data should begin at \(0.\) Unless the reason for beginning at some nonzero value is highlighted and explained, such a chart is more likely to be misleading than informative. We should notice if we are ever shown a bar chart that does not start at \(0\), as it typically means the differences between the heights are being exaggerated. Even starting at \(0\) with non-ratio data is misleading, as multiplicative relationships are inherently meaningless. For this reason, we recommend avoiding bar charts for non-ratio data.
Consider the graph below that shows the number of people playing card games on the Yahoo website on a Sunday and Wednesday in the Spring of \(2001.\) Here, we have two distributions of game frequencies based on the day. Bar graphs are helpful when comparing distributions taking on the same classes. What can we conclude from such a graph?
- Answer
-
The number of people playing Pinochle was the same on these two days. In contrast, about twice as many people were playing hearts on Wednesday as on Sunday. Blackjack was the only game with more players on Sunday than on Wednesday. Facts like these emerge clearly from a well-designed bar graph. The bars are oriented horizontally rather than vertically. The horizontal format is recommended when you have many categories because there is more room for the category labels.
We can also conclude that there were more players overall on Wednesday than on Sunday. This is because Blackjack was the only game with more players on Sunday than Wednesday, and the margin wasn't large enough to offset the rest. Making conclusions about all the categories collectively is not always easy because bar charts are most conducive to comparing parts to parts.
Graphs are pictorial representations of numbers. The following graph, originally from Erickson Times , shows the number of medals per country in the Summer Olympics. Take a minute to observe the graph and see why the pictures in this particular graph may be misleading.
- Answer
-
We naturally expect the representation of the numbers to be proportional to the actual numbers. When looking at Germany's medal count, we see two medals on the graph equal to about \(500\) medals; we reason that each medal pictured would equal about \(250\) medals. France only has \(523\) medals with three units on the graph, and Russia has \(999\) medals with five units on the graph. The numerical order is correct, but there is no consistent, intuitive correspondence between the units on the graph and the actual number of medals. When using pictures, a graph may look fancy, innovative, or visually appealing but can render the graph misleading and ineffective. The purpose of graphical representations is to disseminate information clearly and quickly. Be cautious.


23.4: The Mouth, Pharynx, and Esophagus
By the end of this section, you will be able to:
- Describe the structures of the mouth, including its three accessory digestive organs
- Group the 32 adult teeth according to name, location, and function
- Describe the process of swallowing, including the roles of the tongue, upper esophageal sphincter, and epiglottis
- Trace the pathway food follows from ingestion into the mouth through release into the stomach
In this section, you will examine the anatomy and functions of the three main organs of the upper alimentary canal—the mouth, pharynx, and esophagus—as well as three associated accessory organs—the tongue, salivary glands, and teeth.
The Mouth
The cheeks, tongue, and palate frame the mouth, which is also called the oral cavity (or buccal cavity). The structures of the mouth are illustrated in Figure \(\PageIndex{1}\).
At the entrance to the mouth are the lips, or labia (singular = labium). Their outer covering is skin, which transitions to a mucous membrane in the mouth proper. Lips are very vascular with a thin layer of keratin; hence, the reason they are "red." They have a huge representation on the cerebral cortex, which probably explains the human fascination with kissing! The lips cover the orbicularis oris muscle, which regulates what comes in and goes out of the mouth. The labial frenulum is a midline fold of mucous membrane that attaches the inner surface of each lip to the gum. The cheeks make up the oral cavity’s sidewalls. While their outer covering is skin, their inner covering is mucous membrane. This membrane is made up of non-keratinized, stratified squamous epithelium. Between the skin and mucous membranes are connective tissue and buccinator muscles.  Additionally, notice how these muscles work when you are speaking.
The pocket-like part of the mouth that is framed on the inside by the gums and teeth, and on the outside by the cheeks and lips is called the oral vestibule . Moving farther into the mouth, the opening between the oral cavity and throat (oropharynx) is called the fauces (like the kitchen "faucet"). The main open area of the mouth, or oral cavity proper, runs from the gums and teeth to the fauces.
When you are chewing, you do not find it difficult to breathe simultaneously. The next time you have food in your mouth, notice how the arched shape of the roof of your mouth allows you to handle both digestion and respiration at the same time. This arch is called the palate. The anterior region of the palate serves as a wall (or septum) between the oral and nasal cavities as well as a rigid shelf against which the tongue can push food. It is created by the maxillary and palatine bones of the skull and, given its bony structure, is known as the hard palate. If you run your tongue along the roof of your mouth, you’ll notice that the hard palate ends in the posterior oral cavity, and the tissue becomes fleshier. This part of the palate, known as the soft palate , is composed mainly of skeletal muscle. You can therefore manipulate, subconsciously, the soft palate—for instance, to yawn, swallow, or sing (see Figure \(\PageIndex{1}\)).
A fleshy bead of tissue called the uvula drops down from the center of the posterior edge of the soft palate. Although some have suggested that the uvula is a vestigial organ, it serves an important purpose. When you swallow, the soft palate and uvula move upward, helping to keep foods and liquid from entering the nasal cavity. Unfortunately, it can also contribute to the sound produced by snoring. Two muscular folds extend downward from the soft palate, on either side of the uvula. Toward the front, the palatoglossal arch lies next to the base of the tongue; behind it, the palatopharyngeal arch forms the superior and lateral margins of the fauces. Between these two arches are the palatine tonsils, clusters of lymphoid tissue that protect the pharynx. The lingual tonsils are located at the base of the tongue.
The Tongue
Perhaps you have heard it said that the tongue is the strongest muscle in the body. Those who stake this claim cite its strength proportionate to its size. Although it is difficult to quantify the relative strength of different muscles, it remains indisputable that the tongue is a workhorse, facilitating ingestion, mechanical digestion, chemical digestion (lingual lipase), sensation (of taste, texture, and temperature of food), swallowing, and vocalization.
The tongue is attached to the mandible, the styloid processes of the temporal bones, and the hyoid bone. The hyoid is unique in that it only distantly/indirectly articulates with other bones. The tongue is positioned over the floor of the oral cavity. A medial septum extends the entire length of the tongue, dividing it into symmetrical halves.
Beneath its mucous membrane covering, each half of the tongue is composed of the same number and type of intrinsic and extrinsic skeletal muscles. The intrinsic muscles (those within the tongue) are the longitudinalis inferior, longitudinalis superior, transversus linguae, and verticalis linguae muscles. These allow you to change the size and shape of your tongue, as well as to stick it out, if you wish. Having such a flexible tongue facilitates both swallowing and speech.
As you learned in your study of the muscular system, the extrinsic muscles of the tongue are the palatoglossus, hyoglossus, styloglossus, and genioglossus muscles. These muscles originate outside the tongue and insert into connective tissues within the tongue. The mylohyoid is responsible for raising the tongue, the hyoglossus pulls it down and back, the styloglossus pulls it up and back, and the genioglossus pulls it forward. Working in concert, these muscles perform three important digestive functions in the mouth: (1) position food for optimal chewing, (2) gather food into a bolus (rounded mass), and (3) position food so it can be swallowed.
The top and sides of the tongue are studded with papillae, extensions of lamina propria of the mucosa, which are covered in stratified squamous epithelium (Figure \(\PageIndex{2}\)). Fungiform papillae, which are mushroom shaped, cover a large area of the tongue; they tend to be larger toward the rear of the tongue and smaller on the tip and sides. In contrast, filiform papillae are long and thin. Fungiform papillae contain taste buds, and filiform papillae have touch receptors that help the tongue move food around in the mouth. The filiform papillae create an abrasive surface that performs mechanically, much like a cat’s rough tongue that is used for grooming. Lingual glands in the lamina propria of the tongue secrete mucus and a watery serous fluid that contains the enzyme lingual lipase , which plays a minor role in breaking down triglycerides but does not begin working until it is activated in the stomach. A fold of mucous membrane on the underside of the tongue, the lingual frenulum , tethers the tongue to the floor of the mouth. People with the congenital anomaly ankyloglossia, also known by the non-medical term “tongue tie,” have a lingual frenulum that is too short or otherwise malformed. Severe ankyloglossia can impair speech and must be corrected with surgery.
The Salivary Glands
Many small salivary glands are housed within the mucous membranes of the mouth and tongue. These minor exocrine glands are constantly secreting saliva, either directly into the oral cavity or indirectly through ducts, even while you sleep. In fact, an average of 1 to 1.5 liters of saliva is secreted each day. Usually just enough saliva is present to moisten the mouth and teeth. Secretion increases when you eat, because saliva is essential to moisten food and initiate the chemical breakdown of carbohydrates. Small amounts of saliva are also secreted by the labial glands in the lips. In addition, the buccal glands in the cheeks, palatal glands in the palate, and lingual glands in the tongue help ensure that all areas of the mouth are supplied with adequate saliva.
The Major Salivary Glands
Outside the oral mucosa are three pairs of major salivary glands, which secrete the majority of saliva into ducts that open into the mouth:
- The submandibular glands , which are in the floor of the mouth, secrete saliva into the mouth through the submandibular ducts.
- The sublingual glands , which lie below the tongue, use the lesser sublingual ducts to secrete saliva into the oral cavity.
- The parotid glands lie between the skin and the masseter muscle, near the ears. They secrete saliva into the mouth through the parotid duct, which is located near the second upper molar tooth (Figure \(\PageIndex{3}\)).
Saliva
Saliva is essentially (99.4 percent) water. The remaining 0.6 percent is a complex mixture of ions, glycoproteins, enzymes, growth factors, and waste products. Perhaps the most important ingredient in saliva from the perspective of digestion is the enzyme salivary amylase , which initiates the breakdown of carbohydrates. Food does not spend enough time in the mouth to allow all the carbohydrates to break down, but salivary amylase continues acting until it is inactivated by stomach acids. Bicarbonate and phosphate ions function as chemical buffers, maintaining saliva at a pH between 6.35 and 6.85. Salivary mucus helps lubricate food, facilitating movement in the mouth, bolus formation, and swallowing. Saliva contains immunoglobulin A, which prevents microbes from penetrating the epithelium, and lysozyme, which makes saliva antimicrobial. Saliva also contains epidermal growth factor, which might have given rise to the adage “a mother’s kiss can heal a wound.”
Each of the major salivary glands secretes a unique formulation of saliva according to its cellular makeup. For example, the parotid glands secrete a watery solution that contains salivary amylase. The submandibular glands have cells similar to those of the parotid glands, as well as mucus-secreting cells. Therefore, saliva secreted by the submandibular glands also contains amylase but in a liquid thickened with mucus. The sublingual glands contain mostly mucous cells, and they secrete the thickest saliva with the least amount of salivary amylase.
Homeostatic Imbalances
The Parotid Glands: Mumps
Infections of the nasal passages and pharynx can attack any salivary gland. The parotid glands are the usual site of infection with the virus that causes mumps (paramyxovirus). Mumps manifests by enlargement and inflammation of the parotid glands, causing a characteristic swelling between the ears and the jaw. Symptoms include fever and throat pain, which can be severe when swallowing acidic substances such as orange juice.
In about one-third of males who are past puberty, mumps also causes testicular inflammation, typically affecting only one testis and rarely resulting in sterility. With the increasing use and effectiveness of mumps vaccines, the incidence of mumps has decreased dramatically. According to the U.S. Centers for Disease Control and Prevention (CDC), the number of mumps cases dropped from more than 150,000 in 1968 to fewer than 1700 in 1993 to only 11 reported cases in 2011.
Regulation of Salivation
The autonomic nervous system regulates salivation (the secretion of saliva). In the absence of food, parasympathetic stimulation keeps saliva flowing at just the right level for comfort as you speak, swallow, sleep, and generally go about life. Over-salivation can occur, for example, if you are stimulated by the smell of food, but that food is not available for you to eat. Drooling is an extreme instance of the overproduction of saliva. During times of stress, such as before speaking in public, sympathetic stimulation takes over, reducing salivation and producing the symptom of dry mouth often associated with anxiety. When you are dehydrated, salivation is reduced, causing the mouth to feel dry and prompting you to take action to quench your thirst.
Salivation can be stimulated by the sight, smell, and taste of food. It can even be stimulated by thinking about food. You might notice whether reading about food and salivation right now has had any effect on your production of saliva.
How does the salivation process work while you are eating? Food contains chemicals that stimulate taste receptors on the tongue, which send impulses to the superior and inferior salivatory nuclei in the brain stem. These two nuclei then send back parasympathetic impulses through fibers in the glossopharyngeal and facial nerves, which stimulate salivation. Even after you swallow food, salivation is increased to cleanse the mouth and to water down and neutralize any irritating chemical remnants, such as that hot sauce in your burrito. Most saliva is swallowed along with food and is reabsorbed, so that fluid is not lost.
The Teeth
The teeth, or dentes (singular = dens), are organs similar to bones that you use to tear, grind, and otherwise mechanically break down food.
Types of Teeth
During the course of your lifetime, you have two sets of teeth (one set of teeth is a dentition ). Your 20 deciduous teeth , or baby teeth, first begin to appear at about 6 months of age. Between approximately age 6 and 12, these teeth are replaced by 32 permanent teeth . Moving from the center of the mouth toward the side, these are as follows (Figure \(\PageIndex{4}\)):
- The eight incisors , four top and four bottom, are the sharp front teeth you use for biting into food.
- The four cuspids (or canines) flank the incisors and have a pointed edge (cusp) to tear up food. These fang-like teeth are superb for piercing tough or fleshy foods.
- Posterior to the cuspids are the eight premolars (or bicuspids), which have an overall flatter shape with two rounded cusps useful for mashing foods.
- The most posterior and largest are the 12 molars , which have several pointed cusps used to crush food so it is ready for swallowing. The third members of each set of three molars, top and bottom, are commonly referred to as the wisdom teeth, because their eruption is commonly delayed until early adulthood. It is not uncommon for wisdom teeth to fail to erupt; that is, they remain impacted. In these cases, the teeth are typically removed by orthodontic surgery.
Anatomy of a Tooth
The teeth are secured in the alveolar processes (sockets) of the maxilla and the mandible. Gingivae (commonly called the gums) are soft tissues that line the alveolar processes and surround the necks of the teeth. Teeth are also held in their sockets by a connective tissue called the periodontal ligament.
The two main parts of a tooth are the crown , which is the portion projecting above the gum line, and the root , which is embedded within the maxilla and mandible. Both parts contain an inner pulp cavity , containing loose connective tissue through which run nerves and blood vessels. The region of the pulp cavity that runs through the root of the tooth is called the root canal. Surrounding the pulp cavity is dentin , a bone-like tissue. In the root of each tooth, the dentin is covered by an even harder bone-like layer called cementum . In the crown of each tooth, the dentin is covered by an outer layer of enamel , the hardest substance in the body (Figure \(\PageIndex{5}\)).
Although enamel protects the underlying dentin and pulp cavity, it is still nonetheless susceptible to mechanical and chemical erosion, or what is known as tooth decay. The most common form, dental caries (cavities) develops when colonies of bacteria feeding on sugars in the mouth release acids that cause soft tissue inflammation and degradation of the calcium crystals of the enamel. The digestive functions of the mouth are summarized in Table \(\PageIndex{1}\).
| Structure | Action | Outcome |
|---|---|---|
| Lips and cheeks | Confine food between teeth |
|
| Salivary glands | Secrete saliva |
|
| Tongue’s extrinsic muscles | Move tongue sideways, and in and out |
|
| Tongue’s intrinsic muscles | Change tongue shape |
|
| Taste buds | Sense food in mouth and sense taste |
|
| Lingual glands | Secrete lingual lipase |
|
| Teeth | Shred and crush food |
|
The Pharynx
The pharynx (throat) is involved in both digestion and respiration. It receives food and air from the mouth, and air from the nasal cavities. When food enters the pharynx, involuntary muscle contractions close off the air passageways.
A short tube of skeletal muscle lined with a mucous membrane, the pharynx runs from the posterior oral and nasal cavities to the opening of the esophagus and larynx. It has three subdivisions. The most superior, the nasopharynx, is involved only in breathing and speech. The other two subdivisions, the oropharynx and the laryngopharynx , are used for both breathing and digestion. The oropharynx begins inferior to the nasopharynx and is continuous below with the laryngopharynx (Figure \(\PageIndex{6}\)). The inferior border of the laryngopharynx connects to the esophagus, whereas the anterior portion connects to the larynx, allowing air to flow into the bronchial tree.
Histologically, the wall of the oropharynx is similar to that of the oral cavity. The mucosa includes a stratified squamous epithelium that is endowed with mucus-producing glands. During swallowing, the elevator skeletal muscles of the pharynx contract, raising and expanding the pharynx to receive the bolus of food. Once received, these muscles relax and the constrictor muscles of the pharynx contract, forcing the bolus into the esophagus and initiating peristalsis.
Usually during swallowing, the soft palate and uvula rise reflexively to close off the entrance to the nasopharynx. At the same time, the larynx is pulled superiorly and the cartilaginous epiglottis, its most superior structure, folds inferiorly, covering the glottis (the opening to the larynx); this process effectively blocks access to the trachea and bronchi. When the food “goes down the wrong way,” it goes into the trachea. When food enters the trachea, the reaction is to cough, which usually forces the food up and out of the trachea, and back into the pharynx.
The Esophagus
The esophagus is a muscular tube that connects the pharynx to the stomach. It is approximately 25.4 cm (10 in) in length, located posterior to the trachea, and remains in a collapsed form when not engaged in swallowing. As you can see in Figure \(\PageIndex{7}\), the esophagus runs a mainly straight route through the mediastinum of the thorax. To enter the abdomen, the esophagus penetrates the diaphragm through an opening called the esophageal hiatus.
Passage of Food through the Esophagus
The upper esophageal sphincter , which is continuous with the inferior pharyngeal constrictor, controls the movement of food from the pharynx into the esophagus. The upper two-thirds of the esophagus consists of both smooth and skeletal muscle fibers, with the latter fading out in the bottom third of the esophagus. Rhythmic waves of peristalsis, which begin in the upper esophagus, propel the bolus of food toward the stomach. Meanwhile, secretions from the esophageal mucosa lubricate the esophagus and food. Food passes from the esophagus into the stomach at the lower esophageal sphincter (also called the gastroesophageal or cardiac sphincter). Recall that sphincters are muscles that surround tubes and serve as valves, closing the tube when the sphincters contract and opening it when they relax. The lower esophageal sphincter relaxes to let food pass into the stomach, and then contracts to prevent stomach acids from backing up into the esophagus. Surrounding this sphincter is the muscular diaphragm, which helps close off the sphincter when no food is being swallowed. When the lower esophageal sphincter does not completely close, the stomach’s contents can reflux (that is, back up into the esophagus), causing heartburn or gastroesophageal reflux disease (GERD).
Histology of the Esophagus
The mucosa of the esophagus is made up of an epithelial lining that contains non-keratinized, stratified squamous epithelium, with a layer of basal and parabasal cells. This epithelium protects against erosion from food particles. The mucosa’s lamina propria contains mucus-secreting glands. The muscularis layer changes according to location: In the upper third of the esophagus, the muscularis is skeletal muscle. In the middle third, it is both skeletal and smooth muscle. In the lower third, it is smooth muscle. As mentioned previously, the most superficial layer of the esophagus is called the adventitia, not the serosa. In contrast to the stomach and intestines, the loose connective tissue of the adventitia is not covered by a fold of visceral peritoneum. The digestive functions of the esophagus are identified in Table \(\PageIndex{2}\).
| Action | Outcome |
|---|---|
| Upper esophageal sphincter relaxation | Allows the bolus to move from the laryngopharynx to the esophagus |
| Peristalsis | Propels the bolus through the esophagus |
| Lower esophageal sphincter relaxation | Allows the bolus to move from the esophagus into the stomach and prevents chyme from entering the esophagus |
| Mucus secretion | Lubricates the esophagus, allowing easy passage of the bolus |
Deglutition
Deglutition is another word for swallowing—the movement of food from the mouth to the stomach. The entire process takes about 4 to 8 seconds for solid or semisolid food, and about 1 second for very soft food and liquids. Although this sounds quick and effortless, deglutition is, in fact, a complex process that involves both the skeletal muscle of the tongue and the muscles of the pharynx and esophagus. It is aided by the presence of mucus and saliva. There are three stages in deglutition: the voluntary phase, the pharyngeal phase, and the esophageal phase (Figure \(\PageIndex{8}\)). The autonomic nervous system controls the latter two phases.
The Voluntary Phase
The voluntary phase of deglutition (also known as the oral or buccal phase) is so called because you can control when you swallow food. In this phase, chewing has been completed and swallowing is set in motion. The tongue moves upward and backward against the palate, pushing the bolus to the back of the oral cavity and into the oropharynx. Other muscles keep the mouth closed and prevent food from falling out. At this point, the two involuntary phases of swallowing begin.
The Pharyngeal Phase
In the pharyngeal phase, stimulation of receptors in the oropharynx sends impulses to the deglutition center (a collection of neurons that controls swallowing) in the medulla oblongata. Impulses are then sent back to the uvula and soft palate, causing them to move upward and close off the nasopharynx. The laryngeal muscles also constrict to prevent aspiration of food into the trachea. At this point, deglutition apnea takes place, which means that breathing ceases for a very brief time. Contractions of the pharyngeal constrictor muscles move the bolus through the oropharynx and laryngopharynx. Relaxation of the upper esophageal sphincter then allows food to enter the esophagus.
The Esophageal Phase
The entry of food into the esophagus marks the beginning of the esophageal phase of deglutition and the initiation of peristalsis. As in the previous phase, the complex neuromuscular actions are controlled by the medulla oblongata. Peristalsis propels the bolus through the esophagus and toward the stomach. The circular muscle layer of the muscularis contracts, pinching the esophageal wall and forcing the bolus forward. At the same time, the longitudinal muscle layer of the muscularis also contracts, shortening this area and pushing out its walls to receive the bolus. In this way, a series of contractions keeps moving food toward the stomach. When the bolus nears the stomach, distention of the esophagus initiates a short reflex relaxation of the lower esophageal sphincter that allows the bolus to pass into the stomach. During the esophageal phase, esophageal glands secrete mucus that lubricates the bolus and minimizes friction.
Interactive Link
Watch this animation to see how swallowing is a complex process that involves the nervous system to coordinate the actions of upper respiratory and digestive activities. During which stage of swallowing is there a risk of food entering respiratory pathways and how is this risk blocked?


14: Fundamental Equilibrium Concepts
In this chapter, you will learn how to predict the position of the balance and the yield of a product of a reaction under specific conditions, how to change a reaction's conditions to increase or reduce yield, and how to evaluate an equilibrium system's reaction to disturbances.
Contributors
-
Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).


 You have five choices of sprinkles, four choices of cookie crumbs, five choices of fruit, and three choices of chocolate chunks. For each category of topping, you may choose only one of the available options, or you may choose to skip that category altogether. How many different sundaes could you create out of these choices?
You turn eighteen and your trust fund finally starts paying out. You decide to buy a vehicle, and eventually narrow things down to a choice between five SUVs, four sports cars, and two motorcycles. How many ways are there to choose a vehicle? How many ways are there to choose one vehicle of each type?
- Use the Multiplication Rule to demonstrate that the truth table of a logical statement with \(n\) statement variables requires \(2^n\) rows. That is, demonstrate that there are \(2^n\) different possible combinations of input truth values for \(n\) statement variables.
- How many different truth tables involving \(n\) statement variables exist?
Recall that if \(A\) is a finite set with \(\vert A \vert = n\text{,}\) then \(\vert \mathscr{P}(A) \vert = 2^n\text{.}\) Use the Multiplication Rule to verify this formula by considering the construction of an arbitrary subset of \(A\) as a process of making \(n\) “either-or” decisions.
It is the year 2030, and Alberta has succeeded in seceding from Canada and has become the landlocked Kingdom of Albertania. The King decrees that the kingdom's citizens will all be assigned a hexadecimal ID. That is, using alphabet
\begin{equation*} \Sigma = \{ 0,1,2,3,4,5,6,7,8,9, \mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d},\mathrm{e},\mathrm{f} \}, \end{equation*}
IDs will be words from \(\Sigma ^{\ast}\text{.}\) However, the king is vain and doesn't want any such ID to contain his initials, \(\mathrm{jk}\text{.}\)
For each \(n \ge 1\text{,}\) let \(s_n\) represent number of allowable IDs of length \(n\text{.}\)
- Compute \(s_1\text{,}\) \(s_2\text{,}\) and \(s_3\text{.}\)
- Determine a recurrence relation for \(s_n\) which is valid (at least) for \(n \ge 3\text{.}\)
- Hint.
-
For each allowable word of length \(n - 1\) you can create a word of length \(n\) by adding a new letter onto the end. But you want your new word to also be allowable, so be careful about what you add onto the end!


7.12.2: Formula Review
7.2 Permutations
7.3 Combinations
-
The formula for counting combinations is:
7.5 Basic Concepts of Probability
- For an experiment whose sample space consists of equally likely outcomes, the theoretical probability of the event is the ratio where and denote the number of outcomes in the event and in the sample space, respectively.
7.7 What Are the Odds?
-
For an event
,
- If the odds in favor of are , then .
7.8 The Addition Rule for Probability
-
If
and
are mutually exclusive events, then
.
- If and are events that contain outcomes of a single experiment, then .
7.9 Conditional Probability and the Multiplication Rule
- If and are events associated with the first and second stages of an experiment, then .
7.10 The Binomial Distribution
- Suppose we have a binomial experiment with trials and the probability of success in each trial is . Then:
7.11 Expected Value
- where stands for the sum, meaning we add up the results of the formula that follows over all possible outcomes.


Skip to main content
-
community health workers (CHWs)
-
essential public health workers who link communities, health care systems, and state health departments
-
community-based participatory research (CBPR)
-
collaborative research partnerships between community members, researchers, and public health practitioners
-
digital divide
-
the gap between those who have ready access to and use digital technologies, such as computers, smartphones, and the internet, and those who do not
-
Health in All Policies (HiAP)
-
an approach that seeks to incorporate health considerations into policymaking across various sectors and government agencies
-
internet of medical things (IoMT)
-
integrates medical devices and wearables with internet connectivity for remote monitoring, real-time data collection, and communication between clients and health care providers
-
workforce shortage
-
arises when the availability of skilled workers is inadequate to meet the target audience’s needs


6.4: The Schrӧdinger Equation
-
- Last updated
- Save as PDF
Learning Objectives
By the end of this section, you will be able to:
- Describe the role Schrӧdinger’s equation plays in quantum mechanics
- Explain the difference between time-dependent and -independent Schrӧdinger’s equations
- Interpret the solutions of Schrӧdinger’s equation
In the preceding two sections, we described how to use a quantum mechanical wavefunction and discussed Heisenberg’s uncertainty principle. In this section, we present a complete and formal theory of quantum mechanics that can be used to make predictions. In developing this theory, it is helpful to review the wave theory of light. For a light wave, the electric field \(E(x,t)\) obeys the relation
\[\dfrac{\partial^2E}{\partial x^2} = \dfrac{1}{c^2} \dfrac{\partial^2E}{\partial t^2}, \label{eq1} \]
where \(c\) is the speed of light and the symbol \(∂\) represents a partial derivative . (Recall from Oscillations that a partial derivative is closely related to an ordinary derivative, but involves functions of more than one variable. When taking the partial derivative of a function by a certain variable, all other variables are held constant.) A light wave consists of a very large number of photons, so the quantity \(|E(x,t)|^2\) can interpreted as a probability density of finding a single photon at a particular point in space (for example, on a viewing screen).
There are many solutions to this equation. One solution of particular importance is
\[E(x,t) = A \, \sin \, (kx - \omega t), \label{eq2} \]
where \(A\) is the amplitude of the electric field, \(k\) is the wave number, and \(ω\) is the angular frequency. Combing this equation with Equation \ref{eq1} gives
\[k^2 = \dfrac{\omega^2}{c^2},\label{eq3} \]
According to de Broglie’s equations , we have \(p=ℏk\) and \(E=ℏω\). Substituting these equations into Equation \ref{eq3} gives
\[p = \dfrac{E}{c}, \nonumber \]
or
\[E = pc. \label{eq5} \]
Therefore, according to Einstein’s general energy-momentum equation (Equation 5.10.26 ), Equation \ref{eq5} describes a particle with a zero rest mass . This is consistent with our knowledge of a photon.
This process can be reversed. We can begin with the energy-momentum equation of a particle and then ask what wave equation corresponds to it. The energy-momentum equation of a nonrelativistic particle in one dimension is
\[E = \dfrac{p^2}{2m} + U(x,t), \nonumber \]
where p is the momentum, m is the mass, and U is the potential energy of the particle. The wave equation that goes with it turns out to be a key equation in quantum mechanics, called Schrӧdinger’s time-dependent equation .
THE TIME-DEPENDENT SCHRӦDINGER EQUATION
The equation describing the energy and momentum of a wavefunction is known as the Schrӧdinger equation:
\[-\dfrac{\hbar^2}{2m} \dfrac{\partial^2 \Psi \, (x,t)}{\partial x^2} + U \, (x,t) \, \Psi \, (x,t) = i \hbar \dfrac{\partial \Psi \, (x,t)}{\partial t}. \label{SchroDep} \]
As described in Potential Energy and Conservation of Energy , the force on the particle described by this equation is given by
\[F = - \dfrac{\partial U \, (x,t)}{\partial x}. \label{7.24} \]
This equation plays a role in quantum mechanics similar to Newton’s second law in classical mechanics. Once the potential energy of a particle is specified—or, equivalently, once the force on the particle is specified—we can solve this differential equation for the wavefunction. The solution to Newton’s second law equation (also a differential equation) in one dimension is a function x ( t ) that specifies where an object is at any time t . The solution to Schrӧdinger’s time-dependent equation provides a tool—the wavefunction—that can be used to determine where the particle is likely to be. This equation can be also written in two or three dimensions. Solving Schrӧdinger’s time-dependent equation often requires the aid of a computer.
Consider the special case of a free particle. A free particle experiences no force (\(F = 0\)).Based on Equation \ref{7.24}, this requires only that
\[U \, (x,t) = U_0 = constant. \label{7.25} \]
For simplicity, we set \(U_0 = 0\). Schrӧdinger’s equation then reduces to
\[-\dfrac{\hbar^2}{2m} \dfrac{\partial^2 \Psi \, (x,t)}{\partial x^2} = i \hbar \dfrac{\partial \Psi \, (x,t)}{\partial t}.\label{7.26} \]
A valid solution to this equation is
\[\Psi \, (x,t) = Ae^{i(kx - \omega t)}.\label{7.27} \]
Not surprisingly, this solution contains an imaginary number (\(i = \sqrt{-1}\)) because the differential equation itself contains an imaginary number. As stressed before, however, quantum-mechanical predictions depend only on \(|\Psi \, (x,t)|^2\), which yields completely real values. Notice that the real plane-wave solutions, \(\Psi \, (x,t) = A \, sin \, (kx - \omega t)\) and \(\Psi \, (x,t) = A \, cos \, (kx - \omega t)\), do not obey Schrödinger’s equation. The temptation to think that a wavefunction can be seen, touched, and felt in nature is eliminated by the appearance of an imaginary number. In Schrӧdinger’s theory of quantum mechanics, the wavefunction is merely a tool for calculating things.
If the potential energy function ( U ) does not depend on time, it is possible to show that
\[\Psi \, (x,t) = \psi (x) \, e^{-i\omega t} \label{7.28} \]
satisfies Schrӧdinger’s time-dependent equation, where \(\psi (x)\) is a time -independent function and e−iωte−iωt is a space -independent function. In other words, the wavefunction is separable into two parts: a space-only part and a time-only part. The factor \(e^{-i\omega t}\) is sometimes referred to as a time-modulation factor since it modifies the space-only function. According to de Broglie, the energy of a matter wave is given by \(E = \hbar \omega\), where E is its total energy. Thus, the above equation can also be written as
\[\Psi \, (x,t) = \psi (x) \, e^{-iEt/\hbar}. \label{stationary} \]
Any linear combination of such states (mixed state of energy or momentum) is also valid solution to this equation. Such states can, for example, describe a localized particle (see Figure 7.3.1 )
Exercise \(\PageIndex{1}\)
A particle with mass m is moving along the x -axis in a potential given by the potential energy function \(U(x) = 0.5 m \, \omega^2x^2\). Compute the product \(\Psi \, (x,t)^* U(x) \, \Psi \, (x,t)\). Express your answer in terms of the time-independent wavefunction, \(\psi (x)\).
Answer:
\(0.5 \, m\omega^2 x^2 \, \psi (x)^* \psi(x)\)
Combining Equation \ref{stationary} and Equation \ref{SchroDep}, Schrödinger’s time-dependent equation reduces to the Schrӧdinger’s time-independent equation .
THE TIME-INDEPENDENT SCHRӦDINGER EQUATION
\[- \dfrac{\hbar^2}{2m} \dfrac{d^2 \psi(x)}{dx^2} + U \, (x) \, \psi (x) = E \, \psi(x), \label{SchroIndep} \]
where \(E\) is the total energy of the particle (a real number).
Notice that we use “big psi” (\(\Psi\)) for the time-dependent wavefunction and “little psi” (\(\psi\)) for the time-independent wavefunction. The wave-function solution to this equation must be multiplied by the time-modulation factor to obtain the time-dependent wavefunction.
In the next sections, we solve Schrӧdinger’s time-independent equation for three cases: a quantum particle in a box, a simple harmonic oscillator, and a quantum barrier. These cases provide important lessons that can be used to solve more complicated systems. The time-independent wavefunction \(\psi(x)\) solutions must satisfy three conditions:
- \(\psi (x)\) must be a continuous function.
- The first derivative of \(\psi(x)\) with respect to space, \(d\psi (x) /dx\), must be continuous, unless \(V (x) = \infty\).
- \(\psi (x)\) must not diverge (“blow up”) at \(x = \pm \infty\).
The first condition avoids sudden jumps or gaps in the wavefunction. The second condition requires the wavefunction to be smooth at all points, except in special cases. (In a more advanced course on quantum mechanics, for example, potential spikes of infinite depth and height are used to model solids). The third condition requires the wavefunction be normalizable. This third condition follows from Born’s interpretation of quantum mechanics. It ensures that \(|\psi(x)|^2\) is a finite number so we can use it to calculate probabilities.
Exercise \(\PageIndex{2}\)
Which of the following wavefunctions is a valid wave-function solution for Schrӧdinger’s equation?
Answer:
None. The first function has a discontinuity; the second curve is not even a function - it is double-valued; and the third function diverges so is not normalizable.


12.1: Elonis v. US 575 US____ (2015)
NOTE: Where it is feasible, a syllabus (headnote) will be released, as is being done in connection with this case, at the time the opinion is issued.The syllabus constitutes no part of the opinion of the Court but has been prepared by the Reporter of Decisions for the convenience of the reader.See United States v. Detroit Timber & Lumber Co., 200 U. S. 321 .
SUPREME COURT OF THE UNITED STATES
elonis v . united states
certiorari to the united states court of appeals for the third circuit
No. 13–983. Argued December 1, 2014—Decided June 1, 2015
After his wife left him, petitioner Anthony Douglas Elonis, under the pseudonym “Tone Dougie,” used the social networking Web siteFacebook to post self-styled rap lyrics containing graphically violent language and imagery concerning his wife, co-workers, a kindergarten class, and state and federal law enforcement. These posts were often interspersed with disclaimers that the lyrics were “fictitious” and not intended to depict real persons, and with statements that Elonis was exercising his First Amendment rights. Many who knew him saw his posts as threatening, however, including his boss, who fired him for threatening co-workers, and his wife, who sought and was granted a state court protection-from-abuse order against him.
When Elonis’s former employer informed the Federal Bureau of Investigation of the posts, the agency began monitoring Elonis’s Face-book activity and eventually arrested him. He was charged with five counts of violating 18 U. S. C. §875(c), which makes it a federal crime to transmit in interstate commerce “any communication containing any threat . . . to injure the person of another.” At trial, Elonis requested a jury instruction that the Government was required to prove that he intended to communicate a “true threat.” Instead, the District Court told the jury that Elonis could be found guilty if a reasonable person would foresee that his statements would be interpreted as a threat. Elonis was convicted on four of the five counts and renewed his jury instruction challenge on appeal. The Third Circuit affirmed, holding that Section 875(c) requires only the intent to communicate words that the defendant understands, and that a reasonable person would view as a threat.
Held : The Third Circuit’s instruction, requiring only negligence with respect to the communication of a threat, is not sufficient to support a conviction under Section 875(c). Pp. 7–17.
(a) Section 875(c) does not indicate whether the defendant must intend that the communication contain a threat, and the parties can show no indication of a particular mental state requirement in the statute’s text. Elonis claims that the word “threat,” by definition, conveys the intent to inflict harm. But common definitions of “threat” speak to what the statement conveys—not to the author’s mental state. The Government argues that the express “intent to extort” requirements in neighboring Sections 875(b) and (d) should preclude courts from implying an unexpressed “intent to threaten” requirement in Section 875(c). The most that can be concluded from such a comparison, however, is that Congress did not mean to confine Section 875(c) to crimes of extortion, not that it meant to exclude a mental state requirement. Pp. 7–9.
(b) The Court does not regard “mere omission from a criminal enactment of any mention of criminal intent” as dispensing with such a requirement. Morissette v. United States , 342 U. S. 246 . This rule of construction reflects the basic principle that “wrongdoing must be conscious to be criminal,” and that a defendant must be “blameworthy in mind” before he can be found guilty. Id., at 252. The “general rule” is that a guilty mind is “a necessary element in the indictment and proof of every crime.” United States v. Balint , 258 U. S. 250 . Thus, criminal statutes are generally interpreted “to include broadly applicable scienter requirements, even where the statute . . . does not contain them.” United States v. X-Citement Video, Inc. , 513 U. S. 64 . This does not mean that a defendant must know that his conduct is illegal, but a defendant must have knowledge of “the facts that make his conduct fit the definition of the offense.” Staples v. United States , 511 U. S. 600 , n. 3. Federal criminal statutes that are silent on the required mental state should be read to include “only that mens rea which is necessary to separate” wrongful from innocent conduct. Carter v. United States , 530 U. S. 255 . In some cases, a general requirement that a defendant act knowingly is sufficient, but where such a requirement “would fail to protect the innocent actor,” the statute “would need to be read to require . . . specific intent.” Ibid. Pp. 9–13.
(c) The “presumption in favor of a scienter requirement should apply to each of the statutory elements that criminalize otherwise innocent conduct.” X-Citement Video , 513 U. S. , at 72. In the context of Section 875(c), that requires proof that a communication was transmitted and that it contained a threat. And because “the crucial element separating legal innocence from wrongful conduct,” id., at 73, is the threatening nature of the communication, the mental state requirement must apply to the fact that the communication contains a threat. Elonis’s conviction was premised solely on how his posts would be viewed by a reasonable person, a standard feature of civil liability in tort law inconsistent with the conventional criminal conduct requirement of “awareness of some wrongdoing,” Staples , 511 U. S., at 606–607. This Court “ha[s] long been reluctant to infer that a negligence standard was intended in criminal statutes.” Rogers v. United States , 422 U. S. 35 (Marshall, J., concurring). And the Government fails to show that the instructions in this case required more than a mental state of negligence. Hamling v. United States , 418 U. S. 87 , distinguished. Section 875(c)’s mental state requirement is satisfied if the defendant transmits a communication for the purpose of issuing a threat or with knowledge that the communication will be viewed as a threat. The Court declines to address whether a mental state of recklessness would also suffice. Given the disposition here, it is unnecessary to consider any First Amendment issues. Pp. 13–17.
730 F. 3d. 321, reversed and remanded.
Roberts, C. J., delivered the opinion of the Court, in which Scalia, Kennedy, Ginsburg, Breyer, Sotomayor, and Kagan, JJ., joined. Alito, J., filed an opinion concurring in part and dissenting in part. Thomas, J., filed a dissenting opinion.
Contributors and Attributions
- Authored by : US Government. Located at : https://supreme.justia.com/cases/federal/us/575/13-983/ . License : Public Domain: No Known Copyright


9.4: Positioning in Bed
By the end of this section, you will be able to:
- Describe how improper positions can lead to musculoskeletal complications
- Recognize different devices used for joint support, pressure relief, or proper alignment
- Demonstrate common positions for patients who are immobile
Similar to safe patient transfers and ambulation, the position of patients in bed is also important in maintaining healthy body systems and preventing complications from musculoskeletal injuries and skin breakdown. This section will review patient positioning to prevent complications from immobility, as well as common devices used to aid in patient positioning and body alignment.
Positioning to Prevent Musculoskeletal Complications
An patient who is immobile in bed for long periods of time is at risk for musculoskeletal complications due to reductions in muscle mass and bone mineral density, improper body alignment, and impairment of other body systems. Prolonged time in bed has shown to increase recovery time for patients and impact long-term outcomes. Proper positioning can aid in preventing some of these complications, such as external rotation of the hip and foot drop.
Positioning to Prevent External Rotation of the Hip
Special instructions may be given to patients after surgery to assist in recovery and prevent pain and injury. Patients have just had hip surgery should prevent their legs from abduction (rotating out and away from the body), causing external rotation of the hip, to promote proper alignment and healing. The patient’s provider and physical therapist will make recommendations for the length of time for recovery and restrictions. Typical positioning restrictions include avoiding active extension at the hip and external rotation. In bed, patients should avoid sleeping on the affected side and should have an abductor pillow between their legs, which keeps the hips in a neutral position to avoid external rotation away from the body, and adduction (movement close to the midline of the body). While seated, patients should not bend over (Figure 9.22) or raise their feet to prevent hip hyperflexion and external rotation.
Positioning to Prevent Foot Drop
When lying supine in bed, feet naturally fall into plantar flexion because of gravity. If this position is maintained for long periods of time, damage to the muscles in the foot can cause foot drop syndrome. A condition called foot drop syndrome is the inability to raise the front part of the foot due to weakness and paralysis of the dorsiflexors that allow the foot to lift and maintain itself in a perpendicular position; foot drop syndrome makes walking difficult (Figure 9.23). Foot drop syndrome can affect one or both feet, and can be temporary or permanent. Patients are at a high risk for foot drop syndrome if they are on bedrest for extended periods of time, or have compressive nerve disorders, traumatic injuries, or neurologic disorders. To prevent foot drop syndrome, orthotics designed to keep the foot at a 90-degree angle should be worn while in bed.
One of the most common orthotics used for foot drop is a podus boot . The brace covers the ankle and foot to support the muscles, immobilizes joints, and corrects positioning.
Positioning to Prevent Skin Integrity Complications
Immobility can lead to skin breakdown and the development of pressure injuries. A pressure injury is defined as any localized damage to the skin as well as underlying soft tissue, usually occurring over a bony prominence or related to a medical device (The Joint Commission, 2022). Common areas where pressure injuries develop include the coccyx, heels, ears, and other bony prominences (Figure 9.24). Within a few hours, they can start developing in patients who are bedbound, so it is important to minimize pressure, friction, and shear. The nares of the nose is another location. Oxygen tubing can cause pressure injuries here if a patient is immobile and unable to move the tubing.
The repositioning step, which is a change in position of the patient, either passive or assisted, reduces or relieves pressure in areas prone to injury during long periods of pressure. Repositioning can be done with the use of equipment, pillows, wedges, or other assistive aides to relieve pressure.
Interventions to prevent pressure injury include identifying patients who are at risk for injury as soon as possible. This can be done using a validated assessment tool, such as the Braden Scale, at the beginning of each shift and as the patient’s condition changes. Interventions include repositioning at least every two hours, alternating between the sides and back, and using cushions to help alleviate pressure. Ensure that any medical devices, such as IV or catheter tubing, are not under the patient or pressing into the skin. It is also important to complete a skin assessment with every position change.
The heels can be vulnerable to skin breakdown with constant pressure from the bed. Using pillows to float the heels or using splint devices is an option to reduce pressure. One example is a heel protector , which is designed to adapt to the patient’s foot with adjustable Velcro straps. The material inside the boot-like device is soft and evenly distributes weight to alleviate pressure.
Learn more about the Braden Scale and how it is used to determine a patient’s risk for skin breakdown.
Specialty Mattress
To prevent skin breakdown and muscle wasting in patients who are immobile, specialty mattresses can be used for patient comfort. A specialty mattress is a mattress designed to relieve pressure over bony prominences and prevent injury. Alternating pressure mattresses are designed to reduce friction and shearing and relieve and distribute pressure. Weight is either distributed over a large contact area or mechanically distributed in active therapy. Some common materials, which are low-tech and do not provide adjustable pressure, include air, foam, or gel. Hi-tech mattresses are pressure-adjustable and can be inflated or deflated.
A gel overlay mattress is made with a combination of foam and gel, and conform to the patient’s body and retain less body heat. A foam mattress is made of crosscut foam, which allows parts of the mattress to shift along with the patient’s body. A constant low-level pressure is placed on the body until repositioning occurs. A low air loss mattress delivers active therapy by pushing air through tiny holes on the surface of the mattress to keep the skin dry, wick moisture, and accommodate pressure points. It is attached to an air pump or compressor that maintains inflation. An alternating pressure mattress is powered with air and can be programmed to inflate and deflate certain sections in a programmable cycle. These mattresses are commonly used when a pressure injury is present.
Devices Used for Joint Support or Proper Alignment
Proper alignment is defined as joints, tendons, ligaments, and muscles in line with the pull of gravity. This should be true with lying, sitting, or standing and should not cause excessive stain. Along with the assisted devices that were mentioned in the mobility section, wedge pillows, towels, washcloths, and pillows can also aid in alignment. The body should be aligned so that the spine is straight, head is neutral, and extremities are in functional positions.
Wedge Pillows
A wedge pillow is a large triangular pillows made of foam that elevate different parts of the body (Figure 9.25). Wedges can be used to elevate limbs at an angle, relieve pressure off certain points, or keep the torso supported. Most commonly in the hospital setting, wedge pillows are used behind the patient’s back to maintain a side-lying position.
Towels and Washcloths
Towels and washcloths are easily accessible in most facilities and are easy to use when positioning patients. They serve as adaptable aids in repositioning patients by providing targeted support and alignment. Rolled towels can be strategically placed under specific body parts to alleviate pressure points, support limbs and joints, and maintain proper spinal alignment. For example, a nurse may place a rolled washcloth under the palm of a patient’s contracted hand to wick moisture and maintain a more neutral position of the fingers.
Pillows
Pillows can also aid in elevating extremities and providing support. For patients in a side-lying position, placing a pillow between the knees can prevent pressure injury. Pillows can also be used by the patient to splint the abdomen when doing cough/deep-breathing exercises. Additionally, specialized pressure-relief pillows can help distribute weight evenly, mitigating the risk of pressure ulcers in patients who are bedridden. With their soft and pliable nature, pillows offer not only physical support but also psychological comfort, creating a conducive environment for rest and recovery.
Splints
A splint is a firm support that can be made of metal, plastic, or plaster. They are often used to immobilize or provide extra support to an extremity to promote healing. Splints maintain the extremity in a neutral position to prevent or treat shortening of the muscles and connective tissue.
Common Positions for Immobile Patients
Generally, patients who are immobile and bedbound require repositioning at least every two hours. This reduces the risk of pressure injury , helps blood circulation, and allows frequent skin assessments. Often, diagrams are placed to help nurses and ancillary staff determine the time patients need to turn (Figure 9.26).
Pillows, wedges, sheets, and towels can help align patients in a comfortable position. Common positions include supine, Fowler’s, orthopneic (tripod), prone (stomach), lateral (side-lying), and Sim’s (semiprone) (Table 9.6).
| Position | Description | Illustration |
|---|---|---|
| Supine | Patient lies flat on the back with a pillow for head support. | |
| Fowler’s | Patient lies supine but with the head of the bed raised. | |
| Orthopneic (tripod) | Patient is in a sitting position, leaning on an overbed table. | |
| Prone | Patient lies on the abdomen. | |
| Lateral (side-lying) | Patient lies on the side with the top leg flexed for support. | |
| Sim’s (semiprone) | Patient lies in a position between prone and lateral positions. |


22.2: Basics
a finite subset of a given set
the number of combinations of size \(k\) taken from a set of size \(n\)
alternative notation choices for \(C(n, k)\)
the function \((n,k) \mapsto C(n, k)\)
What are the domain and codomain of the choose function?
Permutations and combinations are different. A permutation is a bijection from a set to itself. Given a fixed chosen ordering of the set elements in a list (considered as inputs), a permutation is essentially an re-ordering of the set elements into a second list, to line up outputs with inputs. So order matters in a permutation . On the other hand, a combination is just a set, and order does not matter in a set, only membership matters. That is, two listings of some of the elements from a set are the same combination if all the same elements are listed, regardless of the order of the elements in the two lists. So order does not matter in a combination .
We have
\begin{equation*} C(n, k) = \dfrac{P(n, k)}{k!} = \dfrac{n!}{k! (n-k)!} \text{.} \end{equation*}
- Proof.
-
Suppose \(\vert A \vert = n\text{.}\) Using the solution to Worked Example 22.1.1 as a model for our proof, we note that each ordered list of \(k\) elements taken from \(A\) defines a combination from \(A\text{,}\) but different orderings of the same \(k\) elements yield the same combination. Define two permutations to be “equivalent” if they are orderings of the same elements, so that equivalent permutations are associated to the same combination. Since there are \(k!\) elements in each equivalence class of permutations, we may apply the Division Rule to obtain
\begin{align*} C(n, k) & = \# \{ \text{combinations} \} \\ & = \dfrac{\# \{ \text{permutations} \}}{\# \{ \text{equivalent permutations in each class} \}} \\ & = \dfrac{P(n, k)}{k!} \text{.} \end{align*}
Finally, to obtain the rightmost formula in the statement of the theorem, we just need to combine the above formula relating \(C(n, k)\) and \(P(n, k)\) with the formula for \(P(n, k)\) from Theorem 21.4.1 .
We also have \(C(n, n-k) = C(n, k)\text{.}\)
- Proof.
-
Calculate
\begin{equation*} C(n, n-k) = \dfrac{n!}{(n-k)!(n-(n-k))!} = \dfrac{n!}{(n-k)! k!} = C(n, k)\text{.} \end{equation*}
Interpret this last corollary as follows: from a set of \(n\) objects, choosing to include \(k\) elements in a combination is equivalent to choosing \(n-k\) objects to reject.
How many double-scoop ice cream combinations are possible if the local ice cream shop features thirty-one different flavours? ( Note: Only flavour combinations are relevant, not which flavour goes on the cone first.)
Solution
From thirty-one flavours, there are
\begin{equation*} C(31, 2) = \dfrac{31!}{2!29!} = \dfrac{31 \cdot 30}{2} = 31 \cdot 15 = 465 \end{equation*}
possibilities for double-scoop cones with two distinct flavours. However, there are an additional \(31\) possibilities for double-scoop cones with two scoops the same flavour. So the answer is
\begin{equation*} 465 + 31 = 496 \text{.} \end{equation*}
How many different colour patterns can we achieve by placing three red bottles and five blue bottles on a shelf? (Assume the bottles are indistinguishable except by colour.)
Solution
There are \(8\) possible positions in which to place a bottle. To create an arbitrary colour pattern, we can choose \(3\) of the positions to be filled by red bottles, then place blue in the remaining positions. So the answer is
\begin{equation*} C(8, 3) = \dfrac{8!}{3!5!} = 56 \text{.} \end{equation*}
Compare the above solution for Worked Example \(\PageIndex{2}\) with the solution for the identical problem in Worked Example 21.3.4 .
How many ways are there to choose a team of five people from a pool of six first-year students and four senior students if the team must have three first-years and two seniors?
Solution
Choose the seniors for team, then the first-years (or vice-versa). Applying the Multiplication Rule to these independent, consecutive tasks yields answer
\begin{equation*} C(4, 2)\cdot C(6, 3) = \left( \dfrac{4!}{2! 2!} \right) \left( \dfrac{6!}{3! 3!} \right) = 120\text{.} \end{equation*}
How many ways are there to choose three teams of four members each from a pool of twenty people, where no person can be on more than one team?
Solution
Solution. 1
Choose the first team (\(C(20, 4)\) ways), then the second team (\(C(16, 4)\) ways), then the third team (\(C(12, 4)\) ways). Applying the
Multiplication Rule
yields a total of
\begin{equation*} C(20, 4) \cdot C(16, 4) \cdot C(12, 4) = \left( \dfrac{20!}{4! \, \cancel{16!}} \right) \left( \dfrac{\cancel{16!}}{4! \, \bcancel{12!}} \right) \left( \dfrac{\bcancel{12!}}{4! \, 8!} \right) = \dfrac{20!}{8! \, (4!)^3}. \end{equation*}
possible teams. However, the way in which we have constructed our teams has imposed an order on the collection of teams (first, second, and third team), when there is no reason to assume such structure. Given a collection of teams, re-ordering the teams themselves (not the people within each team) produces an equivalent collection of teams by membership. As there are \(3!\) ways to reorder the three teams, applying the
Division Rule
gives us a final answer of
\begin{equation*} \dfrac{20!}{3! \, 8! \, (4!)^3} \text{.} \end{equation*}
Solution. 2
Initially choose the twelve people who will make up the three teams, but without yet assigning anyone to a particular team (\(C(20, 12)\) ways). Then, from this reduced pool of candidates, choose the first team (\(C(12, 4)\) ways) and the second team (\(C(8, 4)\) ways). The third team will now consist of the remaining four people from the twelve initially chosen. The
Multiplication Rule
gives a preliminary total of
\begin{equation*} C(20, 12) \cdot C(12, 4) \cdot C(8, 4) = \left( \dfrac{20!}{\cancel{12!} \, 8!} \right) \left( \dfrac{\cancel{12!}}{4! \, \bcancel{8!}} \right) \left( \dfrac{\bcancel{8!}}{4! \, 4!} \right) = \dfrac{20!}{8! \, (4!)^3}. \end{equation*}
But as in the first solution above, we need to account for the fact that we have artificially ranked teams as first, second, and third. Applying the
Division Rule
gives us a final answer of
\begin{equation*} \dfrac{20!}{3! \, 8! \, (4!)^3}. \end{equation*}


15.5: A Project
The main objective of finite group theory is to classify all possible finite groups up to isomorphism. This problem is very difficult even if we try to classify the groups of order less than or equal to \(60\text{.}\) However, we can break the problem down into several intermediate problems. This is a challenging project that requires a working knowledge of the group theory you have learned up to this point. Even if you do not complete it, it will teach you a great deal about finite groups. You can use Table \(15.21\) as a guide.
\(Table \text { } 15.21\). Numbers of distinct groups \(G\text{,}\) \(|G| \leq 60\)
| Order | Number | Order | Number | Order | Number | Order | Number |
| \(1\) | ? | \(16\) | \(14\) | \(31\) | \(1\) | \(46\) | \(2\) |
| \(2\) | ? | \(17\) | \(1\) | \(32\) | \(51\) | \(47\) | \(1\) |
| \(3\) | ? | \(18\) | ? | \(33\) | \(1\) | \(48\) | \(52\) |
| \(4\) | ? | \(19\) | ? | \(34\) | ? | \(49\) | ? |
| \(5\) | ? | \(20\) | \(5\) | \(35\) | \(1\) | \(50\) | \(5\) |
| \(6\) | ? | \(21\) | ? | \(36\) | \(14\) | \(51\) | ? |
| \(7\) | ? | \(22\) | \(2\) | \(37\) | \(1\) | \(52\) | ? |
| \(8\) | ? | \(23\) | \(1\) | \(38\) | ? | \(53\) | ? |
| \(9\) | ? | \(24\) | ? | \(39\) | \(2\) | \(54\) | \(15\) |
| \(10\) | ? | \(25\) | \(2\) | \(40\) | \(14\) | \(55\) | \(2\) |
| \(11\) | ? | \(26\) | \(2\) | \(41\) | \(1\) | \(56\) | ? |
| \(12\) | \(5\) | \(27\) | \(5\) | \(42\) | ? | \(57\) | \(2\) |
| \(13\) | ? | \(28\) | ? | \(43\) | \(1\) | \(58\) | ? |
| \(14\) | ? | \(29\) | \(1\) | \(44\) | \(4\) | \(59\) | \(1 \) |
| \(15\) | \(1\) | \(30\) | \(4\) | \(45\) | ? | \(60\) | \(13\) |
Find all simple groups \(G\) ( \(|G| \leq 60\)). Do not use the Odd Order Theorem unless you are prepared to prove it.
Find the number of distinct groups \(G\text{,}\) where the order of \(G\) is \(n\) for \(n = 1, \ldots, 60\text{.}\)
Find the actual groups (up to isomorphism) for each \(n\text{.}\)


13.3: Geometric Sequences
- Find the common ratio for a geometric sequence.
- List the terms of a geometric sequence.
- Use a recursive formula for a geometric sequence.
- Use an explicit formula for a geometric sequence.
Many jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Suppose, for example, a recent college graduate finds a position as a sales manager earning an annual salary of \($26,000\). He is promised a \(2\%\) cost of living increase each year. His annual salary in any given year can be found by multiplying his salary from the previous year by \(102\%\). His salary will be \($26,520\) after one year; \($27,050.40\) after two years; \($27,591.41\) after three years; and so on. When a salary increases by a constant rate each year, the salary grows by a constant factor. In this section, we will review sequences that grow in this way.
Finding Common Ratios
The yearly salary values described form a geometric sequence because they change by a constant factor each year. Each term of a geometric sequence increases or decreases by a constant factor called the common ratio . The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.
A geometric sequence is one in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence. The common ratio can be found by dividing any term in the sequence by the previous term. If \(a_1\) is the initial term of a geometric sequence and \(r\) is the common ratio, the sequence will be
\[\{a_1, a_1r,a_1r^2,a_1r^3,...\} .\]
- Divide each term by the previous term.
- Compare the quotients. If they are the same, a common ratio exists and the sequence is geometric.
Is the sequence geometric? If so, find the common ratio.
- \(1\), \(2\), \(4\), \(8\), \(16\),...
- \(48\), \(12\), \(4\), \(2\),...
Solution
Divide each term by the previous term to determine whether a common ratio exists.
-
\(\dfrac{2}{1}=2\) \(\dfrac{4}{2}=2\) \(\dfrac{8}{4}=2\) \(\dfrac{16}{8}=2\)
The sequence is geometric because there is a common ratio. The common ratio is \(2\).
-
\(\dfrac{12}{48}=\dfrac{1}{4}\) \(\dfrac{4}{12}=\dfrac{1}{3}\) \(\dfrac{2}{4}=\dfrac{1}{2}\)
The sequence is not geometric because there is not a common ratio.
Analysis
The graph of each sequence is shown in Figure \(\PageIndex{1}\). It seems from the graphs that both (a) and (b) appear have the form of the graph of an exponential function in this viewing window. However, we know that (a) is geometric and so this interpretation holds, but (b) is not.
Figure \(\PageIndex{1}\)
If you are told that a sequence is geometric, do you have to divide every term by the previous term to find the common ratio?
No. If you know that the sequence is geometric, you can choose any one term in the sequence and divide it by the previous term to find the common ratio.
Is the sequence geometric? If so, find the common ratio.
\(5\), \(10\), \(15\), \(20\),...
- Answer
-
The sequence is not geometric because \(\dfrac{10}{5}≠\dfrac{15}{10}\)
Is the sequence geometric? If so, find the common ratio.
\(100\), \(20\), \(4\), \(\dfrac{4}{5}\),...
- Answer
-
The sequence is geometric. The common ratio is \(\dfrac{1}{5}\)
Writing Terms of Geometric Sequences
Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is \(a_1=−2\) and the common ratio is \(r=4\), we can find subsequent terms by multiplying \(−2⋅4\) to get \(−8\) then multiplying the result \(−8⋅4\) to get \(−32\) and so on.
\[\begin{align*} a_1 &= −2 \\ a_2 &= (−2⋅4)=−8 \\ a_3 &= (−8⋅4)=−32 \\ a_4 &= (−32⋅4)=128 \end{align*}\]
The first four terms are \(\{–2, –8, –32, –128\}\).
- Multiply the initial term, \(a_1\), by the common ratio to find the next term, \(a_2\).
- Repeat the process, using \(a_n=a_2\) to find \(a_3\) and then use \(a_3\) to find \(a_4\), until all four terms have been identified.
- Write the terms separated by commons within brackets.
List the first four terms of the geometric sequence with \(a_1=5\) and \(r=–2\).
Solution
Multiply \(a_1\) by \(−2\) to find \(a_2\). Repeat the process, using \(a_2\) to find \(a_3\), and so on.
\[\begin{align*} a_1 &= 5 \\ a_2 &= −2a_1=−10 \\ a_3 &= −2a_2=20 \\ a_4 &= −2a_3=−40 \end{align*}\]
The first four terms are \(\{5,–10,20,–40\}\).
List the first five terms of the geometric sequence with \(a_1=18\) and \(r=\dfrac{1}{3}\).
- Answer
-
\(\left \{18, 6, 2, \dfrac{2}{3}, \dfrac{2}{9} \right \} \)
Using Recursive Formulas for Geometric Sequences
A recursive formula allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is \(9\). Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.
The recursive formula for a geometric sequence with common ratio r and first term \(a_1\) is
\[a_n=ra_{n−1},\;\;\; n≥2\]
- State the initial term.
- Find the common ratio by dividing any term by the preceding term.
- Substitute the common ratio into the recursive formula for a geometric sequence.
Write a recursive formula for the following geometric sequence.
\(\{6, 9, 13.5, 20.25, ...\} \nonumber\)
Solution
The first term is given as \(6\). The common ratio can be found by dividing the second term by the first term.
\(r=\dfrac{9}{6}=1.5 \nonumber\)
Substitute the common ratio into the recursive formula for geometric sequences and define \(a_1\).
\[\begin{align*} a_n &= ra_{n−1} \\ a_n &= 1.5a_{n−1} \text{ for }n≥2 \\ a_1 &= 6 \end{align*}\]
Analysis
The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure \(\PageIndex{2}\).
Figure \(\PageIndex{2}\)
Do we have to divide the second term by the first term to find the common ratio?
No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.
Write a recursive formula for the following geometric sequence.
\(\{2, \frac{4}{3}, \frac{8}{9}, \frac{16}{27}, ...\}\)
- Answer
-
\(\begin{align*}a_1 &= 2 \\ a_n &= \dfrac{2}{3}a_{n−1} \text{ for }n≥2 \end{align*}\)
Using Explicit Formulas for Geometric Sequences
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.
\[a_n=a_1r^{n−1}\]
Let’s take a look at the sequence \(\{18, 36, 72, 144, 288, ...\}\). This is a geometric sequence with a common ratio of \(2\) and an exponential function with a base of \(2\). An explicit formula for this sequence is
\(a_n=18·2^{n−1}\)
The graph of the sequence is shown in Figure \(\PageIndex{3}\).
Figure \(\PageIndex{3}\)
The \(n^{th}\) term of a geometric sequence is given by the explicit formula :
\[a_n=a_1r^{n−1}\]
Given a geometric sequence with \(a_1=3\) and \(a_4=24\), find \(a_2\).
Solution
The sequence can be written in terms of the initial term and the common ratio \(r\).
\(3\), \(3r\), \(3r^2\), \(3r^3\),...
Find the common ratio using the given fourth term.
\[\begin{align*} a_n&=a_1r^{n-1} \\ a_4&=3r^3 \qquad \text{Write the fourth term of sequence in terms of }\alpha_1 \text{ and } r \\ 24&=3r^3 \qquad \text{Substitute }24 \text{ for }a_4 \\ 8&=r^3 \qquad \text{Divide} \\ r&=2 \qquad \text{Solve for the common ratio} \end{align*}\]
Find the second term by multiplying the first term by the common ratio.
\[\begin{align*} a_2 &= 2 \\ a_1 &= 2(3) \\ & = 6 \end{align*}\]
Analysis
The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.
Given a geometric sequence with \(a_2=4\) and \(2a_3=32\), find \(a_6\).
- Answer
-
\(a_6=16,384\)
Write an explicit formula for the nth term of the following geometric sequence.
\(\{2, 10, 50, 250, ...\}\)
Solution
The first term is \(2\). The common ratio can be found by dividing the second term by the first term.
\(\dfrac{10}{2}=5\)
The common ratio is \(5\). Substitute the common ratio and the first term of the sequence into the formula.
\[\begin{align*}a_n &= a_1r^{(n−1)} \\ a_n &= 2⋅5^{n−1} \end{align*}\]
The graph of this sequence in Figure \(\PageIndex{4}\) shows an exponential pattern.
Figure \(\PageIndex{4}\)
Write an explicit formula for the following geometric sequence.
\(\{–1, 3, –9, 27, ...\}\)
- Answer
-
\(a_n=−{(−3)}^{n−1}\)
Solving Application Problems with Geometric Sequences
In real-world scenarios involving arithmetic sequences, we may need to use an initial term of \(a_0\) instead of \(a_1\). In these problems, we can alter the explicit formula slightly by using the following formula:
\(a_n=a_0r^n\)
In 2013, the number of students in a small school is \(284\). It is estimated that the student population will increase by \(4\%\) each year.
- Write a formula for the student population.
- Estimate the student population in 2020.
Solution
-
The situation can be modeled by a geometric sequence with an initial term of \(284\). The student population will be \(104\%\) of the prior year, so the common ratio is \(1.04\).
Let \(P\) be the student population and \(n\) be the number of years after 2013. Using the explicit formula for a geometric sequence we get
\(P_n =284⋅{1.04}^n\)
-
We can find the number of years since 2013 by subtracting.
\(2020−2013=7\)
We are looking for the population after \(7\) years. We can substitute \(7\) for \(n\) to estimate the population in 2020.
\(P_7=284⋅{1.04}^7≈374\)
The student population will be about \(374\) in 2020.
A business starts a new website. Initially the number of hits is \(293\) due to the curiosity factor. The business estimates the number of hits will increase by \(2.6%\) per week.
- Write a formula for the number of hits.
- Estimate the number of hits in \(5\) weeks.
- Answer a
-
\(P_n = 293⋅1.026a^n\)
- Answer b
-
The number of hits will be about \(333\).
Access these online resources for additional instruction and practice with geometric sequences.
Key Equations
| recursive formula for nth term of a geometric sequence | \(a_n=ra_{n−1}\), \(n≥2\) |
| explicit formula for nth term of a geometric sequence | \(a_n=a_1r^{n−1}\) |
Key Concepts
- A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.
- The constant ratio between two consecutive terms is called the common ratio.
- The common ratio can be found by dividing any term in the sequence by the previous term. See Example \(\PageIndex{1}\).
- The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. See Example \(\PageIndex{2}\) and Example \(\PageIndex{4}\).
- A recursive formula for a geometric sequence with common ratio \(r\) is given by \(a_n=ra_{n–1}\) for \(n≥2\).
- As with any recursive formula, the initial term of the sequence must be given. See Example \(\PageIndex{3}\).
- An explicit formula for a geometric sequence with common ratio \(r\) is given by \(a_n=a_1r^{n–1}\). See Example \(\PageIndex{5}\).
- In application problems, we sometimes alter the explicit formula slightly to \(a_n=a_0r^n\). See Example \(\PageIndex{6}\).
Contributors and Attributions
-
Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus .