Question

The first term of a G.P is$7$, the last term is$448$, and the sum of all terms is$889$, then the common ratio is

• A. $5$
• B. $4$
• C. $3$
• D. $2$

Answer 1

The correct option is D

$2$

The explanation for the correct answer.

Solve for the required value.

$a=7$

The last term $=448$

${\mathit{a}}_{\mathbf{n}}\mathbf{=}\mathit{a}{\mathit{r}}^{\mathbf{n}\mathbf{-}\mathbf{1}}$

$a_n=$the last term.

$r=$the common ratio.

$7r^{n-1}=448$

$r^{n-1}=64$

$\Rightarrow r^n=64r\dots \left(i\right)$

$S_n=889$

${\mathit{S}}_{\mathbf{n}}\mathbf{=}\mathit{a}\mathbf{\times }\frac{\mathbf{(}{\mathbf{r}}^{\mathbf{n}}\mathbf{-}\mathbf{1}\mathbf{)}}{\mathbf{(}\mathbf{r}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{}}$

$S_n=$the sum of $n$terms.

$7\times \frac{(64r-1)}{(r-1)}=889$

$64r–1=127(r-1)$

$64r–1=127r-127$

$63r=126$

$\Rightarrow r=2$

Hence, option(D) is the correct answer.