Single Choice

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$$A^{2} - B^{2}$$
Correct Answer
B$$A^{2} + B^{2}$$
C$$4AB$$
DNone

Solution

$$(x + a)^{n} = ^{n}C_{0} x^{n} + ^{n}C_{1} x^{n - 1}a + ^{n}C_{2} x^{n - 2}a^{2} + ^{n}C_{3} x^{n - 3} a^{3} + ^{n}C_{4}x^{n - 4} a^{4} + .....$$
$$= (^{n}C_{0}x^{n} + ^{n}C_{2}x^{n - 2}a^{2} + ^{n}C_{4} x^{n - 4} a^{4} + .....) + (^{n}C_{1}x^{n - 1}a + ^{n}C_{3} x^{n - 3}a^{3} + ^{n}C_{5} x^{n - 5}a^{5}) + ....$$
$$= A + B$$ .... (1)
Similarly, $$(x - a)^{n} = A - B$$ .... (2)
Multiplying eqns. (1) and (2), we get
$$(x^{2} - a^{2})^{n} = A^{2} - B^{2}$$

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