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 21n(n2+1).
Solution: Here a=1 for all, and d=1,2,3,...,n respectively for the n A.P.'s We have to find sum of their nth terms. ∑Tn=[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+n2−1) =2n[n2+1].