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April 23rd, 2014, 07:02 AM   #1
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From: Rourkela, India

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Arithmetic Progression

If sum of n terms = n²p and sum of m terms = m²p, in an Arithmetic progression, m≠n, prove that sum of p terms = p³.

Well I tried to solve this way.

(n/2){2a+(n-1)d} = n²p
=> ½{2a+(n-1)d} = np — (i)

(m/2){2a+(m-1)d} = m²p
=> ½{2a+(m-1)d} = mp — (ii)

Subtracting equation (i) from (ii),
½{2a+(n-1)d - 2a-d(m-1)} = p(n-m)
=> ½d(n-m) = (n-m)
=> d=2p —(iii)

Substituiting d by 2p in equation (ii)
½{2a+(m-1)2p} = mp
=> a=p — (iv)

sum of p terms = (p/2) {2a+(p-1)d}
= (p/2){2p+(p-1)2p}
= p² + p² -p
= p(2p-1)

I'm unable to solve further.
Please help

Last edited by luke97; April 23rd, 2014 at 07:07 AM.
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April 23rd, 2014, 08:09 AM   #2
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Originally Posted by luke97 View Post
sum of p terms = (p/2) {2a+(p-1)d}
= (p/2){2p+(p-1)2p}
= p² + p² -p
You have expanded the expression incorrectly. The last line does not follow from what goes before.
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April 23rd, 2014, 05:04 PM   #3
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(p - 1)2p = 2p^2 - 2p
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