Question

If the (m + 1)^th, (n + 1)^th and (r + 1)^th terms of an AP are in GP and m, n, r are in HP, show that the ratios of the common difference to the first term in the AP is (−2/n).

Answer 1

Let a be the first term and d be the common difference of the AP. Let x, y, z be the (m + 1)^th, (n + 1)^th and (r + 1)^th terms of the AP. Then x = a + md, y = a + nd and z = a + rd, since x, y, z are in GP. Therefore,

⇒ y² = xz

⇒ (a + nd)² = (a + rd) (a + md)