Let a, a r, a r^2, ............

Question

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

Accepted Answer

$$\[\begin{array}{ll} \sum_{n=0}^{\infty} a r^n=57 & \Rightarrow \frac{a}{1-r}=57 \quad \text{.... (i)} \ \sum_{n=0}^{\infty} a^3 r^{3 n}=9747 & \Rightarrow \frac{a^3}{1-r^3}=9747 \quad \text{.... (ii)} \end{array}$$\]

$$\begin{aligned}
& \frac{\left(1-r^3\right)}{(1-r)^3}=\frac{(57)^3}{9747}=19 \
& \Rightarrow \quad \frac{(1-r)\left(1+r+r^2\right)}{(1-r)^3}=19 \
& \Rightarrow \quad 18 r^2-39 r+18=0 \
& \Rightarrow \quad r=\frac{2}{3}, \frac{3}{2} \text { (rejected) } \
& \therefore \quad a=19 \
& \quad a+18 r \
& \quad=19+12=31
\end{aligned}$$

Solutions · 1