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5. In an Α.Ρ., if pth term. is-and q tern is-, prove that the surn of firstpqterms is(pq +1), where pヂq. |
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Answer» Given:-pth term = 1/q,that is ap= a + (p-1)d = 1/q=>aq + (pq - q)d = 1 --- (1)Similarly, we get:-ap + (pq - p)d = 1 --- (2)From (1) and (2), we get:-aq + (pq - q)d =ap + (pq - p)d=>aq - ap = d[pq - p - pq + q]=>a(q - p) = d(q - p)Therefore, a = dEquation (1) becomes,dq + pqd - dq = 1 d = 1/pqHence a = 1/pqConsider, Spq= (pq/2)[2a + (pq - 1)d] = (pq/2)[2(1/pq) + (pq - 1)(1/pq)] = (1/2)[2 + pq - 1] = (1/2)[pq + 1] |
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