Question

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

Answer 1

Let the G.P. be $T_1,T_2,T_3,T_4,...T_{2n}$
Number of terms $=2n$
According to the given condition,
$T_1+T_2+T_3+...+T_{2n}=5[T_1+T_3+...+T_{2n-1}]\Rightarrow T_1+T_2+T_3+...+T_{2n}-5[T_1+T_3+...+T_{2n-1}]=0\Rightarrow T_2+T_4+...+T_{2n}=4[T_1+T_3+...+T_{2n-1}]$
Let the G.P. be $a,ar,{ar}^2,{ar}^3,...$
$\therefore \frac{a{r}^2({\left(r^2\right)}^n-1)}{r^2-1}=\frac{4\times a({\left(r^2\right)}^n-1)}{r^2-1}\Rightarrow a{r}^2=4a\Rightarrow r=\pm 2$
Thus the common ratio of the G.P. is $\pm 2.$