If in a G.P.

Question

If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the terms occupying odd places, then the common ratio of the G.P. is equal to:

Accepted Answer

Let first term $a$, ratio $r$. Sum of all $64$ terms: $S = \dfrac{a(r^{64}-1)}{r-1}$. The odd-placed terms ($1$st, $3$rd, \ldots , $63$rd --- $32$ terms) form a G.P. with ratio $r^2$: $S_{odd} = \dfrac{a(r^{64}-1)}{r^2-1}$.

$$S = 7S_{odd} \implies \frac{1}{r-1} = \frac{7}{r^2-1} \implies r+1 = 7 \implies r=6.$$

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