Question

How to find the sum of the series whose common difference is in ap?

Answer 1

Let T(n) be the n'th term of the series

S = 1 + 3 + 6 + 10 + 15 ........ T(n-1) + T(n)
S = 1 + 3 + 6 + 10.....................+ T(n-1) + T(n)

Subtract both equations :

0 = 1 + ( 2 + 3 + 4 + 5 ....... T(n) - T(n-1)) - T(n)
T(n) = 1 + 2 + 3 + 4 + 5 ......... n
T(n) = n(n+1)/2

Use the summation operator on T(n) to get the sum of your series i.e.

Sum(T(n)) = {(1/2)Sum(n^2) + Sum(n)}
Sum(n) = n(n+1)/2
Sum(n^2) = n(n+1)(2n+1)/6

@skb

Answer 2

for finding the sum a(a-1)