If in a G.P. of 64 terms, the sum of all the terms is 7 times the...

For the given GP, $n = 64, {S}_{n} = 7{S}_{o}; \left\{\text{where}{S}_{o} \to \text{Sum of odd terms}\right\}$

$⇒ a + ar + {ar}^{2} + {ar}^{3} + . . . + {ar}^{63} = 7\left(a + {ar}^{2} + {ar}^{4} \ldots . . + {ar}^{62}\right)$

$⇒ \dfrac{a\left(1 - {r}^{64}\right)}{1 - r} = \dfrac{7a\left(1 - {r}^{64}\right)}{1 - {r}^{2}}$

$⇒ \left(1 - {r}^{64}\right) = \dfrac{7\left(1 - {r}^{64}\right)}{1 + r}$

$⇒ 1 = \dfrac{7}{1 + r}$

$⇒ 1 + r = 7$

$⇒ r = 6$