Prove that $ (a-b)^{2}, a^{2}+b^{2} $ and $ (a+b)^{2} $ are three consecutive terms of an AP.

To do: 

We have to prove that \( (a-b)^{2}, a^{2}+b^{2} \) and \( (a+b)^{2} \) are three consecutive terms of an AP.

Solution:

If $a, b, c$ are three consecutive terms of an AP then $b-a=c-b$

Here,

$a^{2}+b^{2}-((a-b)^{2})=a^2+b^2-(a^2-2ab+b^2)$

$=a^2-a^2+b^2-b^2+2ab$

$=2ab$

$(a+b)^{2}-(a^2+b^2)=a^2+2ab+b^2-a^2-b^2$

$=2ab$

$a^{2}+b^{2}-((a-b)^{2})=(a+b)^{2}-(a^2+b^2)$

Therefore, \( (a-b)^{2}, a^{2}+b^{2} \) and \( (a+b)^{2} \) are three consecutive terms of an AP.

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Updated on: 10-Oct-2022

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