If 7=5+(1)/(7)(5+ )+17^2(5+2 )+17^3(5+3 )+ , then the value of is :

Question

If $7=5+\frac{1}{7}(5+\alpha )+\frac{1}{{7}^{2}}(5+2\alpha )+\frac{1}{{7}^{3}}(5+3\alpha )+\ldots \ldots \ldots \ldots \infty$, then the value of $\alpha $ is :

Accepted Answer

$$\begin{aligned}
& \text { Let } S=5+\frac{1}{7}(5+\alpha)+\frac{1}{7^2}(5+2 \alpha)+\ldots \
& \frac{1}{7} S=\frac{1}{7}(5)+\frac{1}{7^2}(5+\alpha)+\ldots \infty
\end{aligned}$$

$$\begin{aligned}
&\frac{6}{7}(S)=5+\frac{1}{7} \alpha\left(\frac{1}{1-\frac{1}{7}}\right)\
&6=5+\frac{\alpha}{6} \Rightarrow \alpha=6
\end{aligned}$$

Solutions · 1