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Geometric Progressions — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSGeometric Progressions1 Mark
Question
If $\text{a} = 1 + \text{b} + \text{b}_2 + \text{b}_3 + ...\text{to }\infty,$ then write b in terms of a given that |b|<1.
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Answer

$\text{a} = 1 + \text{b} + \text{b}_2 + \text{b}_3 + ...\text{to }\infty,$$\text{a}=\frac{1}{1-\text{b}}$ $\Big[\text{Since},\text{S}_\infty=\frac{\text{a}}{1-\text{r}}\Big]$
$\text{a}(1-\text{b})=1$
$\text{a}-\text{ab}=1$
$\text{ab}=\text{a}-1$
$​​​​\text{b}=\frac{\text{a}-1}{\text{a}}$

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