If $(a - b)^{2}, (a^{2} + b^{2})$ are the first two terms of an AP, then which of the following will be the next term?

2ab

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(a + b)^{2}

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- 2ab

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- (a + b)^{2}

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Solution

The correct option is B $(a + b)^{2}$
Given,
$a_{1} = (a - b)^{2}$
$a_{1} = a^{2} + b^{2} - 2 a b$

and
$a_{2} = a^{2} + b^{2}$

Now,
$d = a_{2} - a_{1}$
$d = a^{2} + b^{2} - a^{2} - b^{2} + 2 a b$
$\Rightarrow d = 2 a b$

The next term
$a_{3} = a_{2} + d$
$a_{3} = a^{2} + b^{2} + 2 a b = (a + b)^{2}$

$∴ N e x t t e r m (a_{3}) = (a + b)^{2}$

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