Question

The sequence $b_n$ is a geometric progression with $\frac{b_2}{b_5}=\frac{1}{4}$ and $b_2+b_5=216$. Find $b_1$

Answer 1

$\frac{b_2}{b_5}=\frac{1}{4}\Rightarrow \frac{ar}{a{r}^4}=\frac{1}{4}$

Where $a$ is the first term & $r$ is the common ratio.

$\Rightarrow \frac{1}{r^3}=\frac{1}{4}\Rightarrow r^3=4\Rightarrow r=\sqrt[3]{34}=4^{\frac{1}{3}}$

And,$b_2+b_5=216$

$\Rightarrow ar+a{r}^4=216\Rightarrow ar(1+r^3)=216$

Replacing, $r=4^{\frac{1}{3}}$

$\Rightarrow ar[1+{\left(4^{\frac{1}{3}}\right)}^3]=216\Rightarrow ar[1+4]=216\Rightarrow ar=\frac{216}{5}\Rightarrow a=\frac{216}{5r}=\frac{216}{5\sqrt[3]{34}}$

$\therefore$ The first term is $\frac{216}{5\sqrt[3]{34}}$