Solve the following
Question:

If a = 1 + b + b2 + b3 + … to ∞, then write b in terms of a.

Solution:

Here, $a=1, b, b^{2}, b^{3}, \ldots \infty$ form an infinite G.P.

$\therefore \mathrm{S}_{\infty}=\mathrm{a}=1+\mathrm{b}+\mathrm{b}^{2}+\mathrm{b}^{3}+\ldots \infty=\frac{1}{1-\mathrm{b}}$

$\Rightarrow a=\frac{1}{1-b}$

$\Rightarrow 1-b=\frac{1}{a}$

$\Rightarrow \mathrm{b}=1-\frac{1}{\mathrm{a}}$

$\therefore \mathrm{b}=\frac{\mathrm{a}-1}{\mathrm{a}}$