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.

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.

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