Question

If p, q, r, s ∊ N and they are four consecutive terms of an A.P.,then pth qth, rth and sth terms of G.P. are in (a) A.P. (b) G.P. (c) H.P. (d) none of these

Answer 1

Step-by-step explanation:

My try : Since a, b, c, d are in HP, 1/a, 1/b, 1/c and 1/d are in AP.

If the common difference of AP is k, then 1/d = 1/a + 3k

==> k = (1/3)(1/d - 1/a) = (1/3){(a - d)/ad} = (a-d)/3ad

and 1/c = 1/a + 2k = 1/a + 2(a-d)/3ad = [3d + 2a - 2d]/3ad = (2a + d)/3ad ==> c = 3ad/(2a + d)

Further, a, b, c are in HP, ==> b = 2ac/(a+c) So, ab + bc = b(a+c) = 2ac/(a+c) = 2ac

ab + bc + cd = 2ac + cd = c(2a+d)

Substituting for c from step (ii) above, ab + bc + cd = {3ad/(2a+d)}*(2a+d) = 3ad [Proved] I think that this process is quite long , is there any another approach of this question.