Question

How many terms of the $G.P$. $3,\frac{3}{2},\frac{3}{4},...$ are needed to give the sum $\frac{3069}{512}$ ?

• A. $9$
• B. $12$
• C. $8$
• D. $10$

Answer 1

Answer: (D)

Let $n$ be the number of terms needed. Given that
$a=3,r=\frac{1}{2}$ and
$S_n=\frac{3069}{512}$
Now, $S_n=\frac{a\left(1-r^n\right)}{1-r}$
$\Rightarrow \frac{3069}{512}$
$=\frac{3\left(1-\frac{1}{2^n}\right)}{1-\frac{1}{2}}=6\left(1-\frac{1}{2^n}\right)$
$\Rightarrow \frac{3069}{3072}=1-\frac{1}{2^n}$
$\Rightarrow \frac{1}{2^n}=1-\frac{3069}{3072}$
$=\frac{1}{1024}$
or $2^n=1024$
$=2^{10}$,
which gives $n=10$.