If the sum of P terms of an A.P. is q and the sum of q terms is p, then the sum of p + q terms will be

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In the given problem, we are given Sp = q and Sq = p We need to find S p+q Now, as we know, S_n = n/2 [2a + (n-1) d] So, S_p = p/2 [2a + (p - 1) d] q = p/2 [ 2a + ( p - 1)d] 2q = 2ap + p (p-1)d ............(1) Similarly, S_q = q/2 [ 2a + (q-1) d] p = q/2 [2a + (q-1)d] 2p = 2ap + q(q-1)d ...............(2) Subtracting (2) from (1), we get 2q - 2p = 2ap + [p ( p - 1) d ] - 2 aq - [q (q-1)d] 2q - 2 p = 2a (p-q) + [ p (p-1) - q(q-1)]d -2(p-q) = 2a(p - q) + [(p2 - q2) - ( p - q)] -2 = 2a + ( p + q - 1 ) d ................(3) Now, `S_(p+q) = (p+q)/2 [2a + (p+q - 1)d] S_(p+q) = ((p+q))/2 (-2) ........(Using 3) S_(p+q) = - (p+q) Thus, S_(p+q) = - (p+q)`

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