Find three numbers in G.P. whose sum is 65 and whose product is 3375.

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

So, according to the question

\( \dfrac{a}{r} + a + ar = 65\)… equation (1)

\( \dfrac{a}{r} + a + ar = 3375\) … equation (2)

From equation (2) we get,

a³ = 3375

a = 15.

From equation (1) we get,

\(\dfrac{ (a + ar + ar^2)}{r}\) = 65

a + ar + ar² = 65r … equation (3)

Substituting a = 15 in equation (3) we get

15 + 15r + 15r² = 65r

15r² – 50r + 15 = 0… equation (4)

Dividing equation (4) by 5 we get

3r² – 10r + 3 = 0

3r² – 9r – r + 3 = 0

3r(r – 3) – 1(r – 3) = 0

r = 3 or r = \( \dfrac{1}{3}\)

Now, the equation will be

\( \dfrac{15}{3}\), 15, 15×3 or

So the terms are 5, 15, 45 or 45, 15, 5

The three numbers are 5, 15, 45.