Question

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

• A. $PQ$
• B. $2PQ$
• C. $4PQ$
• D. None of these

Answer 1

Answer: (C)

${\left(x+a\right)}^n=\left(t_1+t_3+t_5+...\right)+\left(t_2+t_4+t_6+...\right)$
$\therefore {\left(x+a\right)}^n=P+Q\dots \left(i\right)$
and ${\left(x-a\right)}^n=\left(t_1+t_3+t_5+...\right)-\left(t_2+t_4+t_6+...\right)$
${\left(x-a\right)}^n=P-Q\dots \left(ii\right)$
Squaring and subtracting $\left(ii\right)$ from $\left(i\right)$, we get
${\left(x+a\right)}^{2n}-{\left(x-a\right)}^{2n}$
$={\left(P+Q\right)}^2-{\left(P-Q\right)}^2=4PQ$