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

It is given that the sum of all even number of terms in a G.P. is 5 times the sum of terms occupying odd places.

Consider the G.P. be T 1 ,T 2 ,T 3 ,T 4 ,… ,T 2n .

So the number of terms are 2n.

According to the question,

T 1 +T 2 +T 3 +T 4 +⋯ +T 2n =5[ T 1 +T 3 +T 5 +⋯ +T 2n-1 ] T 1 +T 2 +T 3 +T 4 +⋯ +T 2n −5[ T 1 +T 3 +T 5 +⋯ +T 2n-1 ]=0 T 2 +T 4 +⋯ +T 2n =4[ T 1 +T 3 +T 5 +⋯ +T 2n-1 ]

Consider that the G.P. be a,ar,a r 2 ,…, so,

ar( r n −1 ) ( r−1 ) = 4×a( r n −1 ) ( r−1 ) ar=4a r=4

Therefore, 4 is the common ratio of the G.P.