### If there are $n$ numbers of which one is $\left( 1 - \dfrac{ 1 } { n^5 } \right)$ and all the others are $1's,$ then by how much is the arithmetic mean of these numbers less than $1.$

$\dfrac{ 1 } { n^6 }$

Step by Step Explanation:
1. It is given that there are $n$ numbers of which one is $\left( 1 - \dfrac{ 1 } { n^5 } \right)$ and all the others are $1's.$
Therefore, the numbers are $\left( 1 - \dfrac{ 1 } { n^5 } \right), 1, 1, 1 \ldots$ (where $n$ is the total number of numbers in the series)
2. Out of $n$ numbers one is $\left( 1 - \dfrac{ 1 } { n^5 } \right)$ and remaining $n-1$ numbers are $1.$
Therefore, the sum of $n-1$ numbers is $n-1.$
Now, the sum of all numbers in the series $= n-1 + \left( 1 - \dfrac{ 1 } { n^5 } \right) = n - \dfrac{ 1 } { n^5 }$
3. Now, the arithmetic mean of the numbers $=$
 Sum of the all numbers $n$

$= \dfrac { n - \dfrac{ 1 } { n^5 } } { n }$
$= \dfrac{ n } { n } - \dfrac{ 1 } { n^6 }$
$= 1 - \dfrac{ 1 } { n^6 }$
4. Thus, we can say that the arithmetic mean of these numbers is $\dfrac{ 1 } { n^6 }$ less than $1.$