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The sum of first three terms of a G.P. is 13/12, and their product is – 1. Find the G.P.

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

So, according to the question

a/r + a + ar = 13/12 … equation (1)

a/r × a × ar = -1 … equation (2)
From equation (2) we get,

a3 = -1

a = -1

From equation (1) we get,

(a + ar + ar2)/r = 13/12

12a + 12ar + 12ar2 = 13r … equation (3)

Substituting a = – 1 in equation (3) we get

12( – 1) + 12( – 1)r + 12( – 1)r2 = 13r

12r2 + 25r + 12 = 0

12r2 + 16r + 9r + 12 = 0… equation (4)

4r (3r + 4) + 3(3r + 4) = 0

r = -3/4 or r = -4/3 

Now the equation will be

-1/(-3/4), -1, -1×-3/4 or -1/(-4/3), -1, -1×-4/3

4/3, -1, 3/4 or 3/4, -1, 4/3

∴ The three numbers are 4/3, -1, 3/4 or 3/4, -1, 4/3

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