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, $t_{1} = (a - b)^{2}$
$= a^{2} + b^{2} - 2 a b$
and $t_{2} = a^{2} + b^{2}$
Now,
$d = t_{2} - t_{1} = a^{2} + b^{2} - a^{2} - b^{2} + 2 a b$
$\Rightarrow d = 2 a b$
The next term:
$t_{3} = t_{2} + d = a^{2} + b^{2} + 2 a b ∴ t_{3} = a^{2} + b^{2} + 2 a b = (a + b)^{2}$

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