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Q)

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

$\begin{array}{1 1}2 \\ 4 \\ 8 \\ 6 \end{array}$

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A)
Since it is given that the number of terms is even number,
let the number of terms in the G.P be $2n$
$\therefore$ The G.P. is $a,ar,ar^2,.........a.r^{2n-1}$......(i)
The terms occupying odd places form the sequence
$a,ar^2,ar^4,.......ar^{2n-2}$..........(ii)
which is also a G.P. with $1^{st}$ term $=a$ and common ration $=r^2$
and no. of terms $=n$
It is given that $(a+ar+ar^2+.......ar^{2n-1})=5(a+ar^2+ar^4+........a.r^{2n-2})$
We know that sum of $n$ terms of a G.P. $=a.\large\frac{r^n-1}{r-1}$
$\Rightarrow\:a+ar+ar^2+.......a.r^{2n-1}=a.\large\frac{r^{2n}-1}{r-1}$ and
$a+ar^2+ar^4+........a.r^{2n-2}= a.\large\frac{(r^2)^{n}-1}{r^2-1}$$=a.\large\frac{r^{2n}-1}{(r-1)(r+1)} \Rightarrow\: a.\large\frac{(r)^{2n}-1}{r-1}$$=5\times a.\large\frac{(r)^{2n}-1}{(r-1)(r+1)}$
$\Rightarrow\:1=\large\frac{5}{r+1}$
$\Rightarrow\:r+1=5\:\:or\:\:r=4$
$i.e.,$ Common ratio $=4$