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