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  4. If P be the sum of odd terms and Q that of even terms in the expansion of (x + a)n, then the value of [(x + a)2n - (x - a)2n] equals

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Q. If $P$ be the sum of odd terms and $Q$ that of even terms in the expansion of $(x + a)^n$, then the value of $[(x + a)^{2n} - (x - a)^{2n}]$ equals

Binomial Theorem

Solution:

$\left(x + a\right)^{n} = \left(t_{1} + t_{3} + t_{5} + ...\right) + \left(t_{2} + t_{4} + t_{6} + ...\right)$
$\therefore \left(x + a\right)^{n} = P + Q\quad\ldots\left(i\right)$
and $\left(x - a\right)^{n} = \left(t_{1} + t_{3} + t_{5} + ...\right) - \left(t_{2} + t_{4} + t_{6} + ...\right)$
$\left(x - a\right)^{n} = P - Q\quad \ldots \left(ii\right)$
Squaring and subtracting $\left(ii\right)$ from $\left(i\right)$, we get
$\left(x + a\right)^{2n }- \left(x - a\right)^{2n}$
$= \left(P + Q\right)^{2} - \left(P - Q\right)^{2} = 4PQ$