If pth term of an A.P. is q and its qth term is p, show that its (p + q)th term is zero.

Question

KnowledgeBoat · Accepted Answer

Let the first term be a and common difference be d. Then, Tp = a + (p − 1)d = q …(1) Tq = a + (q − 1)d = p …(2) Subtracting equation (2) from equation (1), we get : ⇒ a + (p − 1)d − [a + (q − 1)d] = q − p ⇒ (p − 1)d − (q − 1)d = q − p ⇒ pd - d - qd + d = q - p ⇒ pd - qd = q - p ⇒ (p − q)d = q − p ⇒ (p − q)d = −(p − q) ⇒ d = ⇒ d = -1. Substituting d = −1 in equation (1), we get : ⇒ a + (p − 1)(−1) = q ⇒ a − p + 1 = q ⇒ a = p + q − 1. Now, the (p + q)th term : Tp + q = a + (p + q − 1)d = (p + q − 1) + (p + q − 1)(−1) = (p + q − 1) − (p + q − 1) = p - p + q - q - 1 + 1 = 0. Hence, the (p + q)th term of the A.P. is zero.

Mathematics

If p^th term of an A.P. is q and its q^th term is p, show that its (p + q)^th term is zero.

AP

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Mathematics

If pth term of an A.P. is q and its qth term is p, show that its (p + q)th term is zero.

AP

Related Questions

Insert one arithmetic mean between 3 and 13.

The angles of a polygon are in A.P. with common difference 5°. If the smallest angle is 120°, find the number of sides of the polygon.

If a, b and c are pth, qth and rth terms of an A.P., prove thata(q - r) + b(r − p) + c(p − q) = 0

If p^th term of an A.P. is q and its q^th term is p, show that its (p + q)^th term is zero.

If a, b and c are p^th, q^th and r^th terms of an A.P., prove that
a(q - r) + b(r − p) + c(p − q) = 0