If Tn=2n^2 +3n then prove that Sn=n(n+1)(4n+11)/6

Answers (1)

 Tn=2n2 +3n

\begin{array}{l} S_{n}=\sum\left(2 n^{2}+3 n\right) \\ \Rightarrow S_{n}=2 \sum n^{2}+3 \sum n \\ \Rightarrow S_{n}=2\left(\frac{n(n+1)(2 n+1)}{6}\right)+3\left(\frac{n(n+1)}{2}\right) \\ \Rightarrow S_{n}=[n(n+1)] \times\left[\frac{2 n+1}{3}+\frac{3}{2}\right] \\ \Rightarrow S_{n}=[n(n+1)] \times\left[\frac{4 n+2+9}{6}\right] \\ \Rightarrow S_{n}=[n(n+1)] \times\left[\frac{4 n+11}{6}\right] \\ \Rightarrow S_{n}=\frac{n(n+1)(4 n+11)}{6} \end{array}

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