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

Let the three numbers be   \( \dfrac{a}{r}\), a, ar

So, according to the question

\( \dfrac{a}{r} + a + ar = 38\) … equation (1)

\( \dfrac{a}{r} + a + ar = 1728\) … equation (2)

From equation (2) we get,

a³ = 1728

a = 12.

From equation (1) we get,

\(\dfrac{ (a + ar + ar^2)}{r}\) = 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 = \( \dfrac{3}{2}\)

Now the equation will be

(\( \dfrac{12}{ \dfrac{3}{2}}\)) = 8 or

So the terms are 8, 12, 18

The three numbers are 8, 12, 18