In a G.P. the product of three consecutive terms is 27 and the sum of the product of two terms taken at a time is 57/2 . Find the three terms.

Solution :

Let the three terms be a/r, a, ar

The product of three consecutive terms = 27

(a/r) ⋅ a ⋅ ar = 27

a3 = 27

a = 3

Sum of the product of two terms taken at a time = 57/2

[(a/r) ⋅ a] + [a ⋅ ar] + [ar ⋅ a/r] = 57/2

a2/r + a2r + a2 = 57/2

a2(1/r + r + 1) = 57/2

9(1 + r + r2)/r = 57/2

18(r2 + r + 1) = 57 r

18r2 + 18r + 18 = 57 r

18r2 + 18r - 57r + 18 = 0

18r2 - 39r + 18 = 0

6r2 - 13r + 6 = 0

(2r - 3)(3r - 2) = 0

r = 3/2 and r = 2/3

First term = a/r = 3/(3/2) = 2

Second term = a = 3 = 3

Third term = ar = 3(3/2) = 9/2

Hence the required three terms are 2, 3, 9/2 (or) 9/2, 3, 2.