In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is ℝ − (a, b), then a² + b² is equal to ___

In a G.P., if the product of the first three terms is 27

Q. In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is ℝ − (a, b), then a² + b² is equal to ____.

Correct Answer: 90

Explanation

Let the three terms of the G.P. be:

a, ar, ar²

Their product is given as:

a · ar · ar² = a³r³ = (ar)³ = 27

⇒ ar = 3

Now the sum of the first three terms is:

S = a + ar + ar²

Substitute a = 3/r:

S = 3/r + 3 + 3r

S = 3(r + 1 + 1/r)

Let r + 1/r = t, where r > 0 or r < 0.

For real r, we know:

r + 1/r ≥ 2 or r + 1/r ≤ −2

Thus,

S ≥ 3(2 + 1) = 9

S ≤ 3(−2 + 1) = −3

Hence, the set of possible values of S is:

ℝ − (−3, 9)

So,

a = −3, b = 9

a² + b² = (−3)² + 9² = 9 + 81 = 90

Hence, the required value is 90.

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