First slide

Geometric progression

Question

The first term of an arithmetic progression is 1 and the sum of the first nine terms is 369. The first and the ninth term of a geometric progression coincide with the first and the ninth term of the arithmetic progression. The value of seventh term of the geometric progression is ________.

Moderate

Solution

$\begin{array}{l} 369 = \frac{9}{2} [2 + (9 - 1) d] \\ \Rightarrow 82 = 2 + 8 d \\ \Rightarrow d = 10 \end{array}$
Now, ar⁸ = a + 8d
$\begin{array}{l} \Rightarrow 1 \times r^{8} = 1 + 8 \times 10 \\ \Rightarrow r^{8} = 81 \\ \Rightarrow r = \sqrt{3} \\ \Rightarrow {ar}^{(7 - 1)} = 1 \times (\sqrt{3})^{6} = 27 \end{array}$

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