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, then the value of the ratio of the common difference to the first term of the AP is

(a)   -2/n

(b)   2/n

(c)  -n/2

(d)   n/2

Answer 1

Correct option  (A)  - 2/n

Let a be the first term and d be the common difference of the given AP. Then as given the (m + 1)^th, (n + 1)^th and (r + 1)^th terms are in GP. So,a + md,a + nd,a + rd are in GP. This gives

(a + md)² = (a + md)(a + rd)

⇒ d/a = 2n - (m + r) = d(mr - n²)

⇒ d/a = 2n - (m + r)/mr - n²    .............(1)

Next, m,n,r in HP. So

n = 2mr/m + r  ...............(2)

From Eqs. (1) and (2)