Question

In a G.P. of even number of terms, the sum of all terms is 5 times the sum of all the terms which are in odd place. The common ratio of the G.P. is

• A. $-\frac{4}{5}$
• B. $\frac{1}{5}$
• C. $\frac{4}{5}$
• D. 4

Answer 1

The correct option is D 4

Let the G.P. be $a,\mathrm{ar},{\mathrm{ar}}^2,.....$ with 2n terms.
Since, the sum of all terms is 5 times the sum of the odd terms.
$\therefore a+\mathrm{ar}+{\mathrm{ar}}^2+.....+{\mathrm{ar}}^{2n-1}=5\left(a+{\mathrm{ar}}^2+{\mathrm{ar}}^4+.....+{\mathrm{ar}}^{2n-2}\right)\Rightarrow \frac{a\left(r^{2n}-1\right)}{r-1}=\frac{5a\left({\left(r^2\right)}^n-1\right)}{r^2-1}\left(\mathrm{As}\mathrm{series}\mathrm{of}\mathrm{odd}\mathrm{terms}\mathrm{will}\mathrm{have}n\mathrm{terms}\mathrm{and}\mathrm{common}\mathrm{ratio}{r}^2\right)\Rightarrow 1=\frac{5}{r+1}\Rightarrow r=4$