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In a given AP if pth term is q and qth term is p them show that the nth term is (p + q – n). ㅤㅤㅤㅤㅤㅤㅤㅤ |
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Answer» -step explanation:Given:In a AP series.pth term of AP is q.QTH term of AP is p.To Show:NTH term of AP is (p + q – N)Solution: Let a be the FIRST term and d be the common difference of given AP.As we know that the an AP series is given by★ a + (n – 1)d ★∴ pth term will be ➟ ap = a + (p – 1)d➟ a + (p – 1)d = q.........i∴ qth term will be➟ aq = a + (q – 1)d➟ a + (q – 1)d = p........iiFrom equation (i) & (ii) we geta + (p – 1)d = qa + (q – 1)d = p- ㅤ-ㅤㅤㅤㅤ-____________________(p – q)d = (q – p)Or➮ d = (q –p)/(p – q)➮ d = – (–q + p)/(p – q)➮ d = – (p – q)/(p – q)➮ d = –1Put the value of d in equation (i) a + (p – 1)–1 = q a + (–p + 1) = q a – p + 1 = q a = p + q –1So we GOTA = first term = p + q – 1d = common difference = –1Now nth term of AP will be aⁿ = a + (n – 1)d aⁿ = p + q – 1 + (n – 1)–1 aⁿ = p + q – 1 + (–n + 1) aⁿ = p + q – 1 – n + 1 aⁿ = p + q – nHence, nth term of AP is (p + q – n). |
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