Question

3 × 1² + 5 ×2² + 7 × 3² + ...

Answer 1

Let \[T_n\] be the nth term of the given series.
Thus, we have: 

\[T_n = \left( 2n + 1 \right) n^2 = 2 n^3 + n^2\]

Now, let \[S_n\] be the sum of n terms of the given series.
Thus, we have: 

\[S_n = \sum^n_{k = 1} T_k\]

\[\Rightarrow S_n = \sum^n_{k = 1} \left( 2 k^3 + k^2 \right)\]

\[ \Rightarrow S_n = {2\sum}^n_{k = 1} k^3 + \sum^n_{k = 1} k^2 \]

\[ \Rightarrow S_n = \left[ \frac{2 n^2 \left( n + 1 \right)^2}{4} + \frac{n\left( n + 1 \right)\left( 2n + 1 \right)}{6} \right]\]

\[ \Rightarrow S_n = \left[ \frac{n^2 \left( n + 1 \right)^2}{2} + \frac{n\left( n + 1 \right)\left( 2n + 1 \right)}{6} \right]\]

\[ \Rightarrow S_n = \frac{n\left( n + 1 \right)}{2}\left[ n\left( n + 1 \right) + \frac{2n + 1}{3} \right]\]

\[ \Rightarrow S_n = \frac{n\left( n + 1 \right)}{2}\left( \frac{3 n^2 + 3n + 2n + 1}{3} \right)\]

\[ \Rightarrow S_n = \frac{n\left( n + 1 \right)}{2}\left( \frac{3 n^2 + 5n + 1}{3} \right)\]

\[ \Rightarrow S_n = \frac{n\left( n + 1 \right)}{6}\left( 3 n^2 + 5n + 1 \right)\]