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

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Solution

(d) 4AB
$If A and B denote respectively the sums of odd terms and even terms in the expansion (x + a)^{n} Then, (x + a)^{n} = A + B . . . (1) (x - a)^{n} = A - B . . . (2) Squaring and subtraction equation (2) from (1) we get (x + a)^{2 n} - (x - a)^{2 n} = {(A + B)}^{2} - {(A - B)}^{2} \Rightarrow (x + a)^{2 n} - (x - a)^{2 n} = 4 A B$

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