Class 11

Math

Algebra

Sequences and Series

Find the sum to n terms of the series :$1×2×3+2×3×4+3×4×5+˙˙˙$

so the nth term is $a_{n}=n(n+1)(n+2)$

$S_{n}=∑a_{n}=∑(n)(n_{2}+3n+2)$

$=∑(n_{3}+3n_{2}+2n)$

$=∑n_{3}+3∑n_{2}+2∑n$

$=4n_{2}(n+1)_{2} +63(n)(n+1)(2n+1) +22n(n+1) $

$=4n(n+1) ⋅[n(n+1)+2(2n+1)+4]$

$=4n(n+1) [n_{2}+5n+6]$

$S_{n}=4n(n+1)(n+2)(n+3) $

answer