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

Answer 1

$$\begin{gathered} a_n=128,S_n=225\mathrm{and}r=2 \\ {a}_n=128 \\ \therefore a{r}^{\left(n-1\right)}=128 \\ \Rightarrow 2^{\left(n-1\right)}a=128 \\ \Rightarrow \frac{2^{n}a}{2}=128 \\ \Rightarrow 2^n=\frac{256}{a}........\left(\mathrm{i}\right) \\ \mathrm{Also},S_n=225 \\ \Rightarrow a\left(\frac{r^n-1}{r-1}\right)=225 \\ \Rightarrow a\left(\frac{2^n-1}{2-1}\right)=225 \\ \Rightarrow a\left(\frac{256}{a}-1\right)=225\left[\mathrm{Using}\left(\mathrm{i}\right)\right] \\ \Rightarrow 256-a=225 \\ \Rightarrow a=256-225 \\ \Rightarrow a=31 \end{gathered}$$

Disclaimer: None of the given options are correct. This solution has been created according to the question given in the book.