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 $(x^2 - a^2)^n$ is

• A. $A^2 - B^2$
• B. $A^2 + B^2$
• C. $4AB$
• D. None

Answer 1

Answer: (A)

$(x+a)^{n}={ }^{n} C_{0} x^{n}+{ }^{n} C_{1} x^{n-1} a+\ldots \ldots \ldots \ldots \ldots$
$=\left({ }^{n} C_{0} x^{n}+{ }^{n} C_{2} x^{n-2} a^{2}+\ldots \ldots \ldots\right)$
$+\left({ }^{n} C_{1} x^{n-1} a+{ }^{n} C_{3} x^{n-3} a^{3}+\ldots \ldots\right)$
Given $: A={ }^{n} C_{0} x^{n}+{ }^{n} C_{2} x^{n-2} a^{2}+\ldots \ldots \ldots \ldots \ldots$
$B={ }^{n} C_{1} x^{n-1} \cdot a+{ }^{n} C_{3} x^{n-3} a^{3}+\ldots \ldots \ldots \ldots$
$\Rightarrow (x+a)^{n}=A+B$
$(x-a)^{n}={ }^{n} C_{0} x^{n}-{ }^{n} C_{1} x^{n-1} a+{ }^{n} C_{2} x^{n-2} a^{2}-\ldots \ldots \ldots .$
$=\left({ }^{n} C_{0} x^{n}+{ }^{n} C_{2} x^{n-2} a^{2}+\ldots \ldots \ldots\right)$
$-\left({ }^{n} C_{1} x^{n-1} a+{ }^{n} C_{3} x^{n-3} a^{3}+\ldots\right)$
$\Rightarrow (x-a)^{n}=A-B$
$\therefore \left(x^{2}-a^{2}\right)^{n}=(x+a)^{n} \cdot(x-a)^{n}$
$=(A+B)(A-B)=A^{2}-B^{2}$