Question

if the sum of first p terms is q and the sum of first q terma is p ,then find the sum of first(p+q) terms​

Answer 1

Step-by-step explanation:

$let \: the \: first \: term \: of \: A.P. \: is \: a \: and \: common \: \\ difference \: is \: d. \\ then \: the \: sum \: of \: first \: p \: terms \: = q \\ \frac{p}{2} (2a + (p - 1) \times d) = q \\ p(2a + (p - 1) \times d) = 2q \\ 2ap + ({p}^{2} - p) \times d = 2q........(1) \\ \\ And \: the \: sum \: of \: first \: q \: terms \: = p \\ \frac{q}{2} (2a + (q - 1) \times d) = p \\ q(2a + (q - 1) \times d) = 2p \\ 2aq + ( {q}^{2} - q) \times d = 2p...........(2) \\ Substracting \: equation(2) \: from \: (1) \\ 2a(p - q) + ( {p}^{2} - p - {q}^{2} + q) \times d = - 2(p - q) \\ 2a(p - q) +( (p + q)(p - q) - (p - q)) \times d = - 2(p - q) \\ 2a + (p + q - 1) \times d = - 2 ........(3) \\ \\ the \: sum \: of \: (p + q) \: terms = \\ = \frac{(p + q)}{2} (2a + (p + q - 1) \times d) \\ from \: equation \: (3) \\ = \frac{(p + q)}{2} \times - 2 \\ = - (p + q) \\ Hope \: it \: will \: help \: you........$