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

1.	Find three numbers in G.P. whose sum is 65 and whose product is 3375.
Answer» Let the three numbers be a/r, a, ar So, according to the question a/r + a + ar = 65 … equation (1) a/r × a × ar = 3375 … equation (2) From equation (2) we get, a³ = 3375 a = 15. From equation (1) we get, (a + ar + ar²)/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 = 1/3 Now, the equation will be 15/3, 15, 15×3 or 15/(1/3), 15, 15×1/3 So the terms are 5, 15, 45 or 45, 15, 5 ∴ The three numbers are 5, 15, 45.

Answer»

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

So, according to the question

a/r + a + ar = 65 … equation (1)

a/r × a × ar = 3375 … equation (2)
From equation (2) we get,

a³ = 3375

a = 15.

From equation (1) we get,

(a + ar + ar²)/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 = 1/3

Now, the equation will be

15/3, 15, 15×3 or

15/(1/3), 15, 15×1/3

So the terms are 5, 15, 45 or 45, 15, 5
∴ The three numbers are 5, 15, 45.

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