First slide

Geometric progression

Question

The number of terms of the G.P. $3, \frac{3}{2}, \frac{3}{4}, . . . . . .$ needed to obtain a sum of $\frac{3069}{512}$ is :

Moderate

A

9
B

10
C

11
D

12

Solution

Here $a = 3, r = 1 / 2 .$
Let the number of terms needed be $n,$ then
$\begin{array}{l} \frac{3069}{512} = S_{n} = \frac{3 (1 - (1 / 2)^{n})}{1 - 1 / 2} \\ \Rightarrow \frac{1023}{1024} = 1 - {(\frac{1}{2})}^{n} \\ \Rightarrow {(\frac{1}{2})}^{n} = \frac{1}{1024} \Rightarrow n = 10 \end{array}$

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