Q)  If the sum of m terms of an A.P. is the same as the sum of its n terms, show that the sum of its (m+n) terms is zero.

Ans: Let

a = First term of the given A.P. and

d = Common difference .




2am + m(m-1)d = 2an + n(n-1)d

2am + m(m-1)d - 2an - n(n-1)d = 0

2am - 2an + m(m-1)d - n(n-1)d = 0




2a+(m+n-1)d=0.                 [(m-n)\neq 0]

Now, S_{m+n}=\frac{m+n}{2}\{2a+(m+n-1)d\}=\frac{m+n}{2}\times0=0

Hence Proved!

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