Solve the following

Question:

If $S_{2}$ and $S_{4}$ denote respectively the sum of the squares and the sum of the fourth powers of first $n$ natural numbers, then $\frac{S_{4}}{S_{2}}=$ __________________ .

Solution:

S2 : Sum of the squares of first n natural numbers.

S: Sum of the fourth powers of first n natural numbers.

To find :- $\frac{S_{4}}{S_{2}}$

Since

$S_{4}=\frac{n(n+1)(2 n+1)\left(3 n^{2}+3 n-1\right)}{30}$

$S_{2}=\frac{n(n+1)(2 n+1)}{6}$

Hence,$\frac{S_{4}}{S_{2}}=\frac{n(n+1)(2 n+1)\left(3 n^{2}+3 n-1\right)}{30 \times n(n+1)(2 n+1)} \times 6$

Hence,$\quad \frac{S_{4}}{S_{2}}=\frac{3 n^{2}+3 n-1}{5}$
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