Question

there are n numbers of which one is (1-1/n^5) and all the others are 1's, then by how much is the arithmetic mean of these numbers less than 1.

Answer 1

$The\ numbers\ are\ :\ 1,\ 1,\ 1,\ ...\ (n-1)\ times,\ and\ 1\ -\ \frac{1}{n^5}. \\ \\ Arithmatic\ mean = \frac{(n - 1) + 1 - \frac{1}{n^5}}{n} = \frac{n^6-1}{n^6} \\ \\ Arithmatic\ mean\ is\ less\ than\ 1\ by:\ \ 1-\frac{n^6-1}{n^6} = \frac{n^6 - n^6 + 1}{n^6} = \frac{1}{n^6}$