Question

how many terms of Geometric progression 3,3/2,3/4..... add to 3069/512?

Answer 1

a = 3
r = (3/2) / 3 = 1/2

let the number of terms whose sum is 3069/512 be n.

$S_n= \frac{a(1-r^n)}{1-r} \\ \\ \Rightarrow \frac{3069}{512} = \frac{3(1-(1/2)^n)}{1-(1/2)} \\ \\ \Rightarrow \frac{3069}{512} = \frac{3(1-(1/2)^n)}{1/2} \\ \\ \Rightarrow \frac{3069}{512 \times 3 \times 2} = 1-\frac{1}{2^n}\\ \\ \Rightarrow \frac{1}{2^n} = 1 - \frac{3069}{3072} = \frac{3}{3072} = \frac{1}{1024} \\ \\ \Rightarrow \frac{1}{2^n} = \frac{1}{2^{10}} \\ \\ \Rightarrow n = 10$

So number of terms is 10.