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 [2024]

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Solution

Q.
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 [2024]

1 7

2 4

3 5

4 6

Ans.
(4)

Sum of 64 terms = 7 $\times$ Sum of odd terms

$\Rightarrow a + a r + a r^{2} + \dots + a r^{63} = 7 (a + a r^{2} + a r^{4} + \dots + a r^{62})$

$\Rightarrow 1 + r + r^{2} + \dots + r^{63} = 7 (1 + r^{2} + r^{4} + \dots + r^{62})$

$\Rightarrow \frac{1 (1 - r^{64})}{1 - r} = \frac{7 \times (1 - {(r^{2})}^{32})}{1 - r^{2}} \Rightarrow \frac{1 - r^{64}}{(1 - r)} = \frac{7 \times (1 - r^{64})}{(1 - r) (1 + r)}$

$\Rightarrow 1 + r = 7 \Rightarrow r = 6$

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