Question

The sum of the series $1.2.3+2.3.4+3.4.5+....$to $n$terms is

• A. $n(n+1)(n+2)$
• B. $(n+1)(n+2)(n+3)$
• C. $\frac{n(n+1)(n+2)(n+3)}{4}$
• D. $\frac{(n+1)(n+2)(n+3)}{4}$

Answer 1

The correct option is C

$\frac{n(n+1)(n+2)(n+3)}{4}$

The explanation for the correct option

So, the $n^{th}$term of the A.P. $=r(r+1)(r+2)$

$$\begin{gathered} =r(r^2+3r+2) \\ =r^3+3r^2+2r \end{gathered}$$

The sum of the series

$$\begin{gathered} S_n=\sum _{r=1}^n{T}_r \\ =\sum _{r=1}^n\left(r^3+3r^2+2r\right) \\ ={\left[\frac{n(n+1)}{2}\right]}^2+3\left[\frac{n(n+1)(2n+1)}{6}\right]+2\left[\frac{n(n+1)}{2}\right] \\ =\frac{n(n+1)(n^2+5n+6)}{4} \\ =\frac{n(n+1)(n+2)(n+3)}{4} \end{gathered}$$

Hence, the correct option is $o\mathrm{ption}\left(\mathrm{C}\right)$.