Class 11

Math

Algebra

Sequences and Series

There are n A.P.'s whose common differences are $1,2,3,$_________n respectively, the first term of each being unity. Prove that sum of their nth terms is $21 n(n_{2}+1)$.

We have to find sum of their nth terms.

$∑T_{n}=[1+(n−1).1]+[1+(n−1).2]+...+[1+(n−1).n]$

$=[1+1+1+...nterms]+(n−1)[1+2+3+...nterms]$

$=n+(n−1)2n [1+n]=2n (2+n_{2}−1)$

$=2n [n_{2}+1]$.