If a, b, c are in G.P., prove that: a(b^2+c^2)=c(a^2+b^2) - Vidyadip

Question

If a, b, c are in G.P., prove that:
$\text{a}(\text{b}^2+\text{c}^2)=\text{c}(\text{a}^2+\text{b}^2)$

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Answer

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

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