<s>[INST] What is the sum of $100$ terms of the sequence $a_{k} = \frac{1}{k} - \frac{1}{k+1}$? In the sequence $a_{1}, a_{2}, a_{3}, ..., a_{100}$, the $k$th term is defined by $$a_{k} = \frac{1}{k} - \frac{1}{k+1}$$ for all integers $k$ from $1$ through $100$. What is the sum of $100$ terms of this sequence? 
The answer given is $\frac{100}{101}$, but I am not sure how.
So far I am have plugged in the values of $k's$ and have the following values $$\frac{1}{2}, \frac{1}{6}, \frac{1}{12}, \frac{1}{12}, \frac{1}{20}, \frac{1}{30},....$$
The numerator makes sense to me as it's just $1$ and $100 \times 1 = 100$, however, I am not sure about the denominator. 
 [/INST] $$a_{100} = \left( \frac11 - \frac12 \right)+\left( \frac12 - \frac13 \right)+
\left( \frac13 - \frac14 \right)+\cdots + \left( \frac1{100} - \frac1{101} \right)
\\= \frac11 + \left( -\frac12+\frac12\right)+\left( -\frac13+\frac13\right)
+\cdots +\left( -\frac1{100}+\frac1{100}\right) -\frac1{101} = 1 - \frac1{101}
$$
</s>