The arithmetic mean of the squares of the first n natural numbers is

Concept:

Sum of the first n natural numbers is given by \(\rm \frac {n(n+1)}{2}\)

Sum of the squares of first n natural numbers is given by \( {\rm{\;}}\frac{{{\rm{n\;}}\left( {{\rm{n\;}} + {\rm{\;}}1} \right){\rm{\;}}\left( {2{\rm{n\;}} + {\rm{\;}}1} \right)}}{6}\)

 Arithmetic mean:  The arithmetic mean is the sum of all the numbers in a data set divided by the quantity of numbers in that set.

Calculation:

We have to find the arithmetic mean of the squares of the first n natural numbers,

As we know Sum of the squares of first n natural numbers is given by \( {\rm{\;}}\frac{{{\rm{n\;}}\left( {{\rm{n\;}} + {\rm{\;}}1} \right){\rm{\;}}\left( {2{\rm{n\;}} + {\rm{\;}}1} \right)}}{6}\)

Now, 

 Arithmetic mean = 

\(\rm \frac{Sum \;of\; the \;squares\; of \;first \;n \;natural\; numbers }{n}\) 

\( = \rm \frac{ {\rm{\;}}\frac{{{\rm{n\;}}\left( {{\rm{n\;}} + {\rm{\;}}1} \right){\rm{\;}}\left( {2{\rm{n\;}} + {\rm{\;}}1} \right)}}{6}}{n}\)

= \(\rm \frac { (n+1)(2n+1)}{6}\)