Question

The sum of the first three terms of a G.P. is 39/10, and their product is 1. Find the common ratio and the terms.

Answer 1

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

So, according to the question

a/r + a + ar = 39/10 … 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 = 39/10

10a + 10ar + 10ar² = 39r … equation (3)

Substituting a = 1 in 3 we get

10(1) + 10(1)r + 10(1)r² = 39r

10r² – 29r + 10 = 0

10r² – 25r – 4r + 10 = 0… equation (4)

5r(2r – 5) – 2(2r – 5) = 0

r = 2/5 or 5/2

so now the equation will be,

1/(2/5), 1, 1 × 2/5 or 1/(5/2), 1, 1 × 5/2

5/2, 1, 2/5 or 2/5, 1, 5/2

∴ The three numbers are 2/5, 1, 5/2