Cinemugam (Kollywood): If S1, S2, S3,....Sm are the sums of n terms of m A.P.’s whose first terms are 1,2, 3,...m and whose common differences are 1, 3, 5,..., (2m -1) respectively, then show that S1 + S2 + S3 +............Sm = (mn/2)(1 + mn)

If S1, S2, S3,....Sm are the sums of n terms of m A.P.’s whose first terms are 1,2, 3,...m and whose common differences are 1, 3, 5,..., (2m -1) respectively, then show that S1 + S2 + S3 +............Sm = (mn/2)(1 + mn)

If S1, S2, S3,....Sm are the sums of n terms of m A.P.’s whose first terms are 1,2, 3,...m and whose common differences are 1, 3, 5,..., (2m -1) respectively, then show that S1 + S2 + S3 +............Sm = (mn/2)(1 + mn)

Solution :

n = [(l-a)/d] + 1

n = [((2mn - n + 1) - (1 + n))/2n] + 1

n = [(2mn - n + 1 - 1 - n)/2n] + 1

n = [(2mn - 2n)/2n] + 1

n = (m - 1) + 1

n = m

S1 + S2 + S3 +............Sm

Hence proved.

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If S1, S2, S3,....Sm are the sums of n terms of m A.P.’s whose first terms are 1,2, 3,...m and whose common differences are 1, 3, 5,..., (2m -1) respectively, then show that S1 + S2 + S3 +............Sm = (mn/2)(1 + mn)