Let a,ar,ar^2, ............

Question

Let $a,ar,a{r}^{2}$, ............ be an infinite G.P. If $\sum _{n=0}^{\infty}a{r}^{n}=57$ and $\sum _{n=0}^{\infty}{a}^{3}{r}^{3n}=9747$, then $a+18r$ is equal to

Accepted Answer

$\begin{gathered} \sum _{\infty}^{n=0}a{r}^{n}=57 \Rightarrow \frac{a}{1-r}=57\, \ \sum _{\infty}^{n=0}{a}^{3}{r}^{3n}=9747 \Rightarrow \frac{{a}^{3}}{1-{r}^{3}}=9747\, \end{gathered}$

$\begin{gathered} \frac{(1-{r}^{3})}{(1-r{)}^{3}}=\frac{(57{)}^{3}}{9747}=19 \ \Rightarrow \, \frac{(1-r)(1+r+{r}^{2})}{(1-r{)}^{3}}=19 \ \Rightarrow \, 18{r}^{2}-39r+18=0 \ \Rightarrow \, r=\frac{2}{3},\frac{3}{2} \ 	herefore \, a=19 \ \, a+18r \ \, =19+12=31 \end{gathered}$

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