QJEEM24Ma9e82: JEE Main 2024 Mathematics - Class 11 JEEM24Ma9e82 Solution: Let upto 10 terms and β = ∑ n = 1 10 n 4 . If , then is equal…

$\begin{gathered}\alpha=1^2+4^2+8^2 \ldots . \\t_n=a^2+b n+c\end{gathered}$

$\begin{aligned}& 1=a+b+c \\& 4=4 a+2 b+c \\& 8=9 a+3 b+c\end{aligned}$

On solving we get, $\mathrm{a}=\frac{1}{2}, \mathrm{~b}=\frac{3}{2}, \mathrm{c}=-1$

$\begin{aligned}& \alpha=\sum_{n=1}^{10}\left(\frac{n^2}{2}+\frac{3 n}{2}-1\right)^2 \\& 4 \alpha=\sum_{n=1}^{10}\left(n^2+3 n-2\right)^2, \beta=\sum_{n=1}^{10} n^4 \\& 4 \alpha-\beta=\sum_{n=1}^{10}\left(6 n^3+5 n^2-12 n+4\right)=55(353)+40\end{aligned}$