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

