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Find three number in G.P. whose sum is 38 and their product is 1728.

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

So, according to the question

a/r + a + ar = 38 … equation (1)

a/r × a × ar = 1728 … equation (2)
From equation (2) we get,

a3 = 1728

a = 12.

From equation (1) we get,

(a + ar + ar2)/r = 38

a + ar + ar2 = 38r … equation (3)

Substituting a = 12 in equation (3) we get

12 + 12r + 12r2 = 38r

12r2 – 26r + 12 = 0… equation (4)

Dividing equation (4) by 2 we get

6r2 – 13r + 6 = 0

6r2 – 9r – 4r + 6 = 0

3r(3r – 3) – 2(3r – 3) = 0

r = 3/2 or r = 2/3

Now the equation will be

12/(3/2) = 8 or

12/(2/3) = 18

So the terms are 8, 12, 18

∴ The three numbers are 8, 12, 18

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