If $a, b, c$ are in continued proportion, prove that
$(a + b + c) (a - b + c) = a^{2} + b^{2} + c^{2}$

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Solution

To Prove : $(a + b + c) (a - b + c) = a^{2} + b^{2} + c^{2}$

Proof : $a, b, c$ are in continued proportion.

$∴$ $\frac{a}{b} = \frac{b}{c} = k$ (let)
$b = c k$
$a = b k = (c k) k$ $= c k^{2}$

L.H.S. $= (c k^{2} + c k + c) (c k^{2} - c k + c)$

$= c^{2} (k^{2} + k + 1) (k^{2} - k + 1)$

$= c^{2} [{(k^{2} + 1)}^{2} - {(k)}^{2}]$

$= c^{2} [k^{4} + 2 k^{2} + 1 - k^{2}]$

$= c^{2} [k^{4} + k^{2} + 1]$

R.H.S. $= c^{2} k^{4} + c^{2} k^{2} + c^{2}$

$= c^{2} [k^{4} + k^{2} + 1]$

L.H.S $=$ R.H.S.

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