The sum of the first p terms of an AP in equal to the sum of the

first q terms.Prove that sum of the first p+q terms of the same

AP is zero.

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- Jul 1st 2011, 02:15 AMblackhat123Help required to solve an Arithmatic progression problem
The sum of the first p terms of an AP in equal to the sum of the

first q terms.Prove that sum of the first p+q terms of the same

AP is zero. - Jul 1st 2011, 03:55 AMsa-ri-ga-maRe: Help required to solve an Arithmatic progression problem
Sum of p terms S1 = (p/2)[2a + (p-1)d]

Sum of q terms S2 = (q/2)[2a + (q-1)d]

Given S1 = S2. Solve the equations and find 2a =

Then put this value in the sum of p+q terms. - Jul 1st 2011, 04:08 AMemakarovRe: Help required to solve an Arithmatic progression problem
Note also that the claim holds if p does not equal q.

- Jul 1st 2011, 04:23 AMblackhat123Re: Help required to solve an Arithmatic progression problem
I got 2a+d=0 from the equation S1=S2.But I didn't get how this will help to deduce the sum s(p+q)=0 where s(p+q)=(p+q)/2[2a+(p+q-1)d]

Please help me - Jul 1st 2011, 04:50 AMAlso sprach ZarathustraRe: Help required to solve an Arithmatic progression problem
- Jul 1st 2011, 04:51 AMIsomorphismRe: Help required to solve an Arithmatic progression problem
- Jul 1st 2011, 05:50 AMblackhat123Re: Help required to solve an Arithmatic progression problem
Ya... agreed.

I did some silly mistakes.Now I got it solved.

Thanks all for your help. - Jul 1st 2011, 07:25 AMArchie MeadeRe: Help required to solve an Arithmatic progression problem
You could also use

for q>p

Then

Therefore the average must be zero, which is also the average of the first and last terms of this progression.

This causes the sum of p+q terms to be zero, since