Question

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

Answer 1

Let the G.P. be T1, T2, T3, T4, … T2n.
Number of terms = 2n
According to the given condition,
T1 + T2 + T3 + …+ T2n = 5 [T1 + T3 + … +T2n-1]
⇒ T1 + T2 + T3 + … + T2n-5 [T1 + T3 + … + T2n-1] = 0
⇒ T2 + T4 + … + T2n = 4 [T1 + T3 + … + T2n-1]
Let the G.P. be a, ar, ar2 , ar3 , …
$∴ \frac{ar (r^n - 1) }{ r - 1 }= \frac{4 × a (r^n - 1) }{ r - 1}$
$⇒ ar = 4a$
$⇒ r = 4a$
Thus, the common ratio of the G.P. is 4.