A G.P. consists of an even number of terms.

Question:

A G.P. 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 G.P.

Solution:

Let there be 2n terms in the given G.P. with the first term being a and the common ratio being r.

According to the question

Sum of all the terms = 5 (Sum of the terms occupying the odd places)

$\Rightarrow a_{1}+a_{2}+\ldots+a_{2 n}=5\left(a_{1}+a_{3}+a_{5}+\ldots+a_{2 n-1}\right)$

$\Rightarrow a+a r+\ldots+a r^{2 n-1}=5\left(a+a r^{2}+\ldots+a r^{2 n-2}\right)$

$\Rightarrow a\left(\frac{1-r^{2 n}}{1-r}\right)=5 a\left\{\frac{1-\left(r^{2}\right)^{n}}{1-r^{2}}\right\}$

$\Rightarrow 1+r=5$

$\therefore r=4$

 

 

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