Question

The sum of first three terms of a G.P. is $16$ and the sum of the next three terms, is $128$. Determine the first term, the common ratio and the sum to n terms of the G.P.

Answer 1

Given,

$a+ar+a{r}^2=16$

$\Rightarrow a(1+r+r^2)=16$.....(1)

$a{r}^3+a{r}^4+a{r}^5=128$

$\Rightarrow a{r}^3(1+r+r^2)=128$......(2)

From (1) and (2), we get,

$\frac{16}{a}=\frac{128}{a{r}^3}$

$\Rightarrow r^3=8$

$\therefore r=2$

From (1)

$a(1+r+r^2)=16$

$a(1+2+2^2)=16$

$a=\frac{16}{7}$

$S_n=\frac{a(r^n-1)}{r-1}$

$=\frac{\frac{16}{7}(2^n-1)}{2-1}$

$=\frac{16}{7}(2^n-1)$