Question:
If A and B are the sums of odd and even terms respectively in the expansion of (x + a)n, then (x + a)2n − (x − a)2n is equal to
(a) 4 (A + B)
(b) 4 (A − B)
(c) AB
(d) 4 AB
Solution:
(d) 4 AB
If $A$ and $B$ denote respectively the sums of odd terms and even terms in the expansion $(x+a)^{n}$
Then, $(x+a)^{n}=A+B \quad \ldots(1)$
$(x-a)^{n}=A-B \quad \ldots(2)$
Squaring and subtraction equation (2) from (1) we get
$(x+a)^{2 n}-(x-a)^{2 n}=(A+B)^{2}-(A-B)^{2}$
$\Rightarrow(x+a)^{2 n}-(x-a)^{2 n}=4 A B$
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