Question

A GP consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places, find the common ratio of the GP.

Answer 1

Let the terms of the G.P. be a, ar, ar² , ar³ , … , ar^n-2 , ar^n-1 

Sum of a G.P. series is represented by the formula, S_n = \(a\frac{r^n - 1}{r-1}\), when r≠1. 

‘Sn’ represents the sum of the G.P. series up to n^th terms, 

‘a’ represents the first term, 

‘r’ represents the common ratio and 

‘n’ represents the number of terms. 

Thus, the sum of this G.P. series is S_n = \(a\frac{r^n - 1}{r-1}\)

The odd terms of this series are a, ar² , ar⁴ , … , ar^n-2 

{Since the number of terms of the G.P. series is even; the 2^nd last term will be an odd term.} 

Here, 

No. of terms will be \(\frac{n}{2}\) as we are splitting up the n terms into 2 equal parts of odd and even terms. {since the no. of terms is even, we have 2 equal groups of odd and even terms } 

Sum of the odd terms ⇒