The nth term of a G.P. is 128 and the sum of its n terms is 225.

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.

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