Question

2 + 10 + 30 + 68 + 130 + ...

Answer 1

Let $T_n$ be the nth term of the given series.

∴ $T_n=n+n^3$

Now, let $S_n$ be the sum of n terms of the given series.

Thus, we have:

$$\begin{gathered} S_n=\sum _{k=1}^n\left(k+k^3\right) \\ \Rightarrow S_n=\sum _{k=1}^{n}k+\sum _{k=1}^n{k}^3 \\ \Rightarrow S_n=\frac{n\left(n+1\right)}{2}+{\left[\frac{n\left(n+1\right)}{2}\right]}^2 \\ \Rightarrow S_n=\frac{n\left(n+1\right)}{2}\left[1+\frac{n\left(n+1\right)}{2}\right] \\ \Rightarrow S_n=\frac{n\left(n+1\right)}{4}\left(n^2+n+2\right) \end{gathered}$$