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Question
The first term of an A.P.is a, and the sum of the first p terms is zero, show that the sum of its next q terms is `(-a(p + q)q)/(p - 1)`
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Solution
Given that a1 = a and Sp = 0
Sum of next q terms of the given A.P. = Sp+q – Sp
∴ `"S"_(p + q) = (p + q)/2[2a + (p + q - 1)d]`
And Sp = `p/2 [2a + (p - 1)d]` = 0
⇒ 2a + (p – 1)d = 0
⇒ (p – 1)d = – 2a
⇒ d = `(-2a)/(p - 1)`
Sum of next q terms = Sp+q – Sp
= `(p + q)/2[2a + (p + q- 1)d]` = 0
= `(p + q)/2[2a + (p + q - 1) ((-2a)/(p - 1))]`
= `(p + q)/2[2a + ((p - 1)(-2a))/(p - 1) - (2aq)/(p - 1)]`
= `(p + q)/2[2a - 2a - (2aq)/(p - 1)]`
= `((p + q))/2((-2aq)/(p - 1))`
= `(-a(p + q)q)/(p - 1)`
Hence, the required sum = `(-a(p + q)q)/(p - 1)`
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