If the sum of odd numbered terms and the sum of even numbered terms in the expansion of

Question:

If the sum of odd numbered terms and the sum of even numbered terms in the expansion of $(x+a)^{n}$ are $A$ and $B$ respectively, then the value of $\left(x^{2}-a^{2}\right)^{n}$ is

(a) $A^{2}-B^{2}$

(b) $A^{2}+B^{2}$

(c) $4 A B$

(d) none of these

Solution:

(a) $A^{2}-B^{2}$

If $A$ and $B$ denote respectively the sums of odd terms and even terms in the expansion $(x+a)^{n}$

Then, $(x+a)^{n}=A+B \quad \ldots(1)$

$(x-a)^{n}=A-B \quad \ldots(2)$

Multplying both the equations we get

$(x+a)^{n}(x-a)^{n}=A^{2}-B^{2}$

$\Rightarrow\left(x^{2}-a^{2}\right)^{n}=A^{2}-B^{2}$

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