Question:
The nth term of a G.P. is 128 and the sum of its n terms is 225. If its common ratio is 2, then its first term is
(a) 1
(b) 3
(c) 8
(d) none of these
Solution:
$a_{n}=128, S_{n}=225$ and $r=2$
$a_{n}=128$
$\therefore a r^{(n-1)}=128$
$\Rightarrow 2^{(n-1)} a=128$
$\Rightarrow \frac{2^{n} a}{2}=128$
$\Rightarrow 2^{n}=\frac{256}{a} \quad \ldots \ldots \ldots(\mathrm{i})$
Also, $S_{n}=225$
$\Rightarrow a\left(\frac{r^{n}-1}{r-1}\right)=225$
$\Rightarrow a\left(\frac{2^{n}-1}{2-1}\right)=225$
$\Rightarrow a\left(\frac{256}{a}-1\right)=225 \quad[\operatorname{Using}(\mathrm{i})]$
$\Rightarrow 256-a=225$
$\Rightarrow a=256-225$
$\Rightarrow a=31$
Disclaimer: None of the given options are correct. This solution has been created according to the question given in the book.