Question

The sum of $n$ terms of a GP is $255$, $n^{th}$ term is $128$ and common ratio is $2$, then the first term will be

• A. $2$
• B. $1$
• C. $3$
• D. $5$

Answer 1

Answer: (B)

$1$

Given that, sum of $n$ terms of a GP is $255$ and common ratio is $2$

i.e. $\frac{a(r^n-1)}{r-1}=255(\because r>1)$ ...(i), where $r$ is common ratio and $a$ is first term

Also, $n^{th}$ term is $128$
i.e. $a{r}^{n-1}=128$ ...(ii)
and $r=2$ ....(iii)
From Equations (i), (ii) and (iii)
$a{\left(2\right)}^{n-1}=128$ ....(iv)
and $\frac{a(2^n-1)}{2-1}=255$ .....(v)
On dividing Eq (v) by Eq (iv), we get
$\frac{2^n-1}{2^{n-1}}=\frac{255}{128}$

$\Rightarrow 2-2^{-n+1}=\frac{255}{128}$
$\Rightarrow 2^{-n}=2^{-8}$

$\Rightarrow n=8$
On putting $n=8$ in Eq (iv), we get

$a.2^7=128=2^7$
$\Rightarrow a=1$