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Geometric Progressions — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSGeometric Progressions3 Marks
Question
If $\frac{1}{\text{a}+\text{b}},\frac{1}{2\text{b}},\frac{1}{\text{b}+\text{c}}$ are three consecutive terms ofan A.P., prove that a, b, c are the three consecutive terms of a G.P.
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Answer

$\frac{1}{\text{a}+\text{b}},\frac{1}{2\text{b}},\frac{1}{\text{b}+\text{c}}\text{ are in A.P.}$ $\frac{2}{2\text{b}}=\frac{1}{(\text{a}+\text{b})}+\frac{1}{(\text{b}+\text{c})}$ $\frac{1}{\text{b}}=\frac{\text{b}+\text{c}+\text{a}+\text{b}}{(\text{a}+\text{b})(\text{b}+\text{c})}$ $\frac{1}{\text{b}}=\frac{2\text{b}+\text{c}+\text{a}}{\text{ab}+\text{ac}+\text{b}^2+\text{bc}}$ $\text{ab}+\text{ac}+\text{b}^2+\text{bc}=2\text{b}^2+\text{bc}+\text{ba}$ $\text{b}^2+\text{ac}=2\text{b}^2$ $\text{b}^2=\text{ac}$ So, a, b, c are in G.P.

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