Question

Find the sum to n terms

$1\times 2+2\times 3+3\times 4+4\times 5+\dots \dots$

Answer 1

The given series is $1\times 2+2\times 3+3\times 4+4\times 5+\dots \dots$ to n terms.

Let $a_n$ be the nth term of the given series

$\therefore a_n=$ [nth term of 1, 2, 3 ....] [nth term of 2, 3, 4, 5 .....]

$=[1+(n-1)\times 1][2+(n-1)\times 1]$

$=n(n+1)=n^2+n$

$\therefore S_n={\sum }_{k=1}^n{a}_k={\sum }_{k=1}^n(k^2+k)$

$=[1^2+1]+[2^2+2]+[3^2+3]+\dots \dots +[n^2+n]$

$=[1^2+2^2+3^2+\dots \dots +n^2]+[1+2+3+\dots \dots +n]$

$=\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}$

$\Rightarrow \frac{n(n+1)}{2}[\frac{2n+1}{3}+1]$

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

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