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) 4 AB
(d) none of these

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Solution

(a) $A^{2} - B^{2}$
$If A and B denote respectively the sums of odd terms and even terms in the expansion (x + a)^{n} Then, (x + a)^{n} = A + B . . . (1) (x - a)^{n} = A - B . . . (2) Multplying both the equations we get (x + a)^{n} (x - a)^{n} = A^{2} - B^{2} \Rightarrow (x^{2} - a^{2})^{n} = A^{2} - B^{2}$

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If the sum of odd numbered terms and the sum of even numbered terms in the expansion of x+an are A and B respectively, then the value of x2-a2n is (a) A2-B2 (b) A2+B2 (c) 4 AB (d) none of these

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