If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eight, tenth and twelfth terms is 15309, then the sum of its first nine terms is : [2025]

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Solution

Q.
If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is : [2025]

1 760

2 755

3 750

4 757

Ans.
(4)

We have, $a r + a r^{3} + a r^{5} = 21, a r^{7} + a r^{9} + a r^{11} = 15309$

$\Rightarrow a r (1 + r^{2} + r^{4}) = 21$ ... (i)

and $a r^{7} (1 + r^{2} + r^{4}) = 15309$ ... (ii)

Divide equation (ii) by (i), we get

$\Rightarrow \frac{a . r^{7}}{a r} = \frac{15309}{21} \Rightarrow r^{6} = 729$

$\Rightarrow r = 3 and a = \frac{7}{91}$

Now, $S_{9} = \frac{a \cdot (r^{9} - 1)}{r - 1}$

$= \frac{\frac{7}{91} (19683 - 1)}{2}$

$= \frac{7 \times 19682}{91 \times 2}$

$= \frac{9841}{13} = 757$ .

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