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Question

If $a, b, c$ are in A.P prove that:
(i) ${(a - c)}^{2} = 4 (a - b) (b - c)$

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Solution

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$b = \frac{a + b}{2} \Rightarrow$
$R H S = 4 (a - \frac{a + c}{2}) (\frac{a + c}{2} - c)$
$= 4 (\frac{2 a - a - c}{2}) (\frac{a + c - 2 c}{2})$
$= \frac{4}{4} (a - c) (a - c)$
$= (a - c)^{2} = R H S$

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