Question

The sum of an infinite geometric series of real numbers is 14, and the sum of the cubes of the terms of this series is 392. The first term of the series is
1. -14
2. 10
3. 7
4. -5

Answer 1

Correct Answer - Option 3 : 7

Concept :

A geometric series is called infinite geometric series only when r < 1.

Sum of Infinite geometric series of form a, ar, ar², ar³,....... = a/(1 - r).

Calculation :

Given that the value of this a/(1 - r) = 14.

⇒ Cubing on both sides, we get a³/(1 - r)³ = 14³.................................equation 1.

⇒ The Sum of cubes of the terms of the above series will be as follows :

  a³, a³r³, a³r⁶, a³r⁹, ..........

⇒ The Sum of the terms of the above infinite series = a3/(1 - r3).

⇒ It is also given that this a3/(1 - r3) = 392...............................equation 2.

⇒ Solving both the equations, we get :

⇒ equation 2 divided by equation 1 :

⇒ 7(1 - r)³ = (1 - r³).

⇒ 6r³ - 21r² + 21r - 6 = 0............................By hit and trial, we get r = 1 and then factorize.

⇒ (r - 1)(r - 2)(6r - 3) = 0.

⇒ r = 1, 2, 1/2.

As it is an infinite series, r should be less than 1, Hence r = 1/2.

⇒ As \(r = {1 \over 2}\) we subsequently get a = 7.