If$b$ is the first term of an infinite G.P. whose sum is five, then $b$lies in the interval :

Solution:

Let $b,br,b{r}^2,\dots \dots$ been infinite G.P. with first term$b$, common ratio$r$ and sum = 5.

$\therefore 5=\frac{b}{1-r};\left|r\right|<1\Rightarrow \frac{b}{5}=1-r\Rightarrow r=1-\frac{b}{5}$

Now, $\left|r\right|<1\Rightarrow \left|1-\frac{b}{5}\right|<1\Rightarrow -1<1-\frac{b}{5}<1$

$\begin{array}{l}\Rightarrow 0<-\frac{b}{5}+2<2\Rightarrow 0<\frac{10-b}{5}<2\\ \Rightarrow 0<-b+10<10\\ \Rightarrow b<10\text{ and }b>0\text{ or }b\in (0,10)\end{array}$