Question

please answer.......If the sums of m A.P.s are given by S1,S2,S3,....Sm and the first terms are 1,2,3,...,m and the differences are 1,3,5,...,(2m-1)then prove that

S1+S2+SE+....+Sm=mn/2(mn+1)



please answer.....hurry

Answer 1

Given common difference d = 1.

We know that sum of n terms = n/2(2a + (n-1) * d)

                   sum of 1st term s1 = n/2(2 * 1 + (n - 1) * 1)

                  sum of 2nd term s1 = n/2(2 * 2 + (n - 1) * 2)

                 sum of mth term sm = m/2(2 * m + (n - 1) * (2m - 1))



Required series will be :


= n/2(2(1 + 2 + 3 + 4 + ... + m)  + (n - 1) (1 + 3 + 5 + ... (2m - 1))

We know that sum of n natural numbers = n(n + 1)/2.

= n/2(2(m + 1)/2) + (n - 1) m/2(2 + (m - 1) * 2)

= mn/2 (mn + 1).

Hope this helps!