If a, b, c, d are in G.P., prove that: ab-cd b^2-c^2 = a+c b - Vidyadip

Question

If a, b, c, d are in G.P., prove that:
$\frac{\text{ab}-\text{cd}}{\text{b}^2-\text{c}^2}=\frac{\text{a}+\text{c}}{\text{b}}$

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Answer

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

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