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

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Solution

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

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