Question

If S₂ and S₄ denote respectively the sum of the squares and the sum of the fourth powers of first n natural numbers, then $\frac{S_4}{S_2}=$_______________.

Answer 1

S₂ : Sum of the squares of first n natural numbers.
S₄ : Sum of the fourth powers of first n natural numbers.
To find :- $\frac{S_4}{S_2}$
$$\begin{gathered} \mathrm{Since} \\ {S}_4=\frac{n\left(n+1\right)\left(2n+1\right)\left(3n^2+3n-1\right)}{30} \\ {S}_2=\frac{n\left(n+1\right)\left(2n+1\right)}{6} \\ \mathrm{Hence},\frac{S_4}{S_2}=\frac{n\left(n+1\right)\left(2n+1\right)\left(3n^2+3n-1\right)}{30\times n\left(n+1\right)\left(2n+1\right)}\times 6 \\ \mathrm{Hence},\frac{S_4}{S_2}=\frac{3n^2+3n-1}{5} \end{gathered}$$