World Languages, asked by brainlyclasher, 8 months ago

In a given AP if the pth term is q and qth term is p ,then show that the nth term is p+q-n​

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Answered by Anonymous
11

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let \:  \: a \:  \: be \: the \: first \: term \: and \: d \: be \: the \: common \: difference \: of \: the \: given \: ap \: \\ then \\ s _{p} = s _{q}  \implies \frac{p}{2} (2a + (p - 1)d) =  \frac{q}{2} (2a + (q - 1)d \\  \implies(p - q)(2a)  = (q - p)(q + p - 1) \\  \implies2a = (1 - p - q)d \:  \:  \:  \:  \:  \:  \: .....(1) \\ sum \: of \: the \: first \: (p + q) \: terms \: of \: the \: given \: ap \\  =  \frac{(p  + q)}{2} (2a + (p + q - 1)d) \\  =  \frac{(p + q)}{2} .(1 - p - q)d + (p + q - 1)d \:  \:  \:  \:  \:  \:  \:  \: (using \: 1) \\   = 0

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