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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 of the G.P. is equal to
Sum of all 64 terms = $$S_{64} = a\frac{r^{64}-1}{r-1}$$.
Sum of odd terms (32 terms, GP with first term a, ratio r²): $$S_{odd} = a\frac{r^{64}-1}{r^2-1}$$.
$$S_{64}/S_{odd} = 7$$. $$\frac{r^2-1}{r-1} \cdot \frac{r-1}{1}$$... Actually $$S_{64}/S_{odd} = \frac{(r^{64}-1)/(r-1)}{(r^{64}-1)/(r^2-1)} = \frac{r^2-1}{r-1} = r+1 = 7$$. So $$r = 6$$.
Option (4).
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