If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio…

Given the sum of all the terms is 7 times the sum of the odd terms of the G.P.

\(a+a r+a r^2+a r^3+\ldots+a r^{63}=7\left(a+a r^2+a r^4 \ldots+a r^{62}\right)\)

\(\Rightarrow \frac{a\left(1-r^{64}\right)}{1-r}=\frac{7a\left(1-\left(r^2\right)^{32}\right)}{1-r^2}\)

\(\Rightarrow \frac{a\left(1-r^{64}\right)}{1-r}=\frac{7 a\left(1-r^{64}\right)}{1-r^2}\)

\(\Rightarrow 1+r=7\\ \Rightarrow r=6\)