Question

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. where p is not equal to q.

Answer 1

Hey friend!!

Here's ur answer..

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Sum of p terms =
$\frac{p}{2} (2a + (p - 1)d)$
Sum of q terms =
$\frac{q}{2} (2a + (q - 1)d)$
Sp = Sq

$\frac{p}{2} (2a + (p - 1)d) = \frac{q}{2} (2a + pd) \\ p(2a + pd - d) = q(2a + pd) \\ p(2a + pd - d) = (p + 1)(2a + pd) \\ 2ap + {p}^{2} d - pd = 2ap + {p}^{2} d + 2a + pd \\ - pd = 2a + pd \\ 2a + 2pd = 0 \\ a + pd = 0 \\ a + (q - 1)d = 0$
Qth term = 0

Sum of (p+q)th term =
$= \frac{p + q}{2} (2a + (p + q - 1)d \\ = \frac{q - 1 + q}{2} (2a + (q - 1 + q - 1)d) \\ = \frac{2q - 1}{2} (2a + (2q - 2)d) \\ = 2q - 1(a + (q - 1)d) \\ = 2q - 1(0) \\ = 0$
Hence, proved...

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Hope it may help you...

Thank you :) :))