Question

Does there exist a GP containing 27, 8 and 12 as three of its terms? If it exists, how many such progressions are possible.

Answer 1

Let 8 be the m^th, 12 be the nth and 27 be the t^th terms of a GP whose first term is A and the common ratio is R. Then

There are infinity of sets of values of m, n, t which satisfy this relation. For example, take m = 1. Then

2 + t/3 = n = k ⇒ n = k, t = 3k - 2

By giving different values to k we get integral values of n and t. Hence, there are an infinite number of GPs whose terms are 27, 8, 12 (may not be consecutive).

From Eqs. (1) and (2), we get