# The 20th term of an arithmetic series is log20 and the 32nd term is log32. Exactly one term in the sequence is a rational number. What is the rational number?

## The answer is $\frac{1}{n + 1}$, but I'm not sure how to get there.

Sep 6, 2016

The tenth term is log10, which equals 1.

#### Explanation:

If the 20th term is log 20, and the 32nd term is log32, then it follows that the tenth term is log10. Log10=1. 1 is a rational number.

When a log is written without a "base" (the subscript after log), a base of 10 is implied. This is known as the "common log". Log base 10 of 10 equals 1, because 10 to the first power is one. A helpful thing to remember is "the answer to a log is the exponent".

A rational number is a number that can be expressed as a ration, or fraction. Note the word RATIO within RATIOnal. One can be expressed as 1/1.

I don't know where the $\frac{1}{n + 1}$ comes from!