Question

In a $G.P$ sum of $n$ terms is $255$, the last term is $128$ and the common ratio is $2$. Find $n$

Answer 1

$1=T_n={ar}^{n-1}=128$, $r=2$
$S_n=\frac{a(r^n-1)}{r-1}=255$
or $\frac{a{r}^{n-1}.r-a}{r-1}=255$ or $\frac{128.2-a}{2-1}=255$
$\therefore 256-255=a$ or $a=1$
Now ${ar}^{n-1}=128$ or $\therefore {1.\left(2\right)}^{n-1}=2^7$
$\therefore n-1=7$ or $n=8$