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Geometric Progressions — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSGeometric Progressions4 Marks
Question
If a, b, c, d are in G.P., prove that: $(\text{a}+\text{b}+\text{c}+\text{d})^2=(\text{a}+\text{b}^2)+2(\text{b}+\text{c})^2+(\text{c}+\text{d})^2$
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Answer

a, b and c are in G.P.$\therefore\text{b}^2=\text{ac }\cdots(1)$
$\text{L.H.S}=({\text{a}+\text{b}+\text{c}+\text{d})^2}$
$=(\text{a}+\text{b})^2+2(\text{a}+\text{b})(\text{c}+\text{d})+(\text{c}+\text{d})^2$
$=(\text{a}+\text{b})^2+2(\text{ac}+\text{ad}+\text{bc}+\text{bd})+(\text{c}+\text{d})^2$
$=(\text{a}+\text{b})^2+2(\text{b}^2+\text{bc}+\text{bc}+\text{c}^2)+(\text{c}+\text{d})^2$ $[\text{Using (1)}]$
$=(\text{a}+\text{b})^2+2(\text{b}+\text{c})^2+(\text{c}+\text{d})^2$
$=\text{R.H.S}$
$\therefore\text{R.H.S}=\text{L.H.S}$

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