Question

if TN=2n res2 +3n then prove that Sn=n(n+1)(4n+11)/6

Answer 1

$T_n=2 n^{2} +3n\\ \\S_n=\sum(2 n^{2} +3n)\\ \\ \Rightarrow S_n=2 \sum n^{2} +3 \sum n\\ \\ \Rightarrow S_n=2( \frac{n(n+1)(2n+1)}{6} ) +3( \frac{n(n+1)}{2} )\\ \\ \Rightarrow S_n=[n(n+1)] \times [ \frac{2n+1}{3}+ \frac{3}{2} ] \\ \\ \Rightarrow S_n=[n(n+1)] \times [ \frac{4n+2+9}{6}] \\ \\ \Rightarrow S_n=[n(n+1)] \times [ \frac{4n+11}{6}] \\ \\ \Rightarrow S_n=\frac{n(n+1)(4n+11)}{6}$