Question

If p times the pth term of an AP is equal to q times the qth term, prove that
its (p + q)th term is zero, provided p ≠ q.

Answer 1

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 = d Equation (1) becomes, dq + pqd - dq = 1   d = 1/pq Hence a = 1/pq Consider, 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]