Question

12. If the sum of first p terms of an A.P. is equal to
the sum of first q terms, then show that the sum
of its first (p+q) terms is zero. p is not equal to q​

Answer 1

Answer: If a is the first term and d the common difference of an AP, then

S(n) = (n/2) [ 2a + (n-1)d.

Given : S(p) = S(q), ... [ where ... p ≠ q.]

∴ (p/2) [ 2a + (p-1)d ] = (q/2) [ 2a + (q-1)d ]

∴ 2ap + p(p-1)d = 2aq + q(q-1)d

∴ ( 2ap - 2aq ) + [ (p²-p) - (q²-q) ]d = 0

∴ 2a(p-q) + [ (p²-q²) - (p-q) ]d = 0

∴ 2a(p-q) + (p-q)[ (p+q) - 1]d = 0

∴ dividing throughout by (p-q),

. . 2a + ( p+q-1)d = 0

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