I know that $\frac{1}{7}$ and $\frac{1}{3}$ are both repeating decimals, meaning they have a pattern of digits that repeat infinitely after the decimal point. To find their decimal equivalents, I can use long division or a calculator. I get $\frac{1}{7} = 0.142857142857\ldots$ and $\frac{1}{3} = 0.333333333333\ldots$. To find their sum, I can align the decimal points and add the digits column by column, carrying over any digits that exceed 9. I get $0.142857142857\ldots + 0.333333333333\ldots = 0.476190476190\ldots$. I notice that the sum also has a repeating pattern of six digits: 476190. To find the 20th digit after the decimal point, I need to figure out which cycle of the pattern it belongs to and where in that cycle it is. I can do this by dividing 20 by 6, the length of the pattern. I get $\frac{20}{6} = 3$ with a remainder of 2. This means that the 20th digit is in the fourth cycle of the pattern, and it is the second digit in that cycle. I can look at the pattern and see that the second digit is 7. Therefore, the 20th digit after the decimal point of the sum is 7.


What is the 20th digit after the decimal point of the sum of the decimal equivalents for the fractions $\frac{1}{7}$ and $\frac{1}{3}$?