If A and B are the sums of odd and even terms respectively in the expansion of

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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