x times the xth term of an AP is equal to y times the yth term of this AP. Which term of this AP is equal to zero.
Explanation:
Let the first term be a and common difference be d.
⇒ Tx = a + (x - 1)d, and ⇒ Ty = a + (y - 1)d
Given, xTx = yTy
⇒ x(a + (x - 1)d) = y(a + (y-1)d) ⇒ ax + x2d - xd = ay + y2d - yd ⇒ a(x - y) + d(x2 - y2) - d(x - y) = 0 ⇒ (x - y)(a + (x + y)d - d) = 0 ⇒ a + (x + y - 1)d = 0 = Tx+y
∴ (x + y)th term is zero.
Hence, option (b).
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