In the expansion of `(x+a)^n` if the sum of odd terms is `P` and the sum of even terms is `Q ,` then (a)`P^2-Q^2=(x^2-a^2)^n` (b)`4P Q=(x+a)^(2n)-(x-a)^(2n)` (c)`2(P^2+Q^2)=(x+a)^(2n)+(x-a)^(2n)` (d)all of these

Options

A. `P^(2) - Q^(2) = (x^(2) - a^(2))^(n)`
B. `4PQ = (x+a)^(2n) - (x-a)^(2n)`
C. `2(P^(2) + Q^(2)) = (x+a)^(2n) + (x-a)^(2n)`
D. none of these

Answer

A, B, C

Explanation

We have
`(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_(n)a^(n)`
`= [.^(n)C_(0)x^(n)+.^(n)C_(2)x^(n-2)+"..."]+[.^(n)C_(1)x^(n-1)a+.^(n)C_(3)x^(m-3)a^(3)+"...."]`
or `(x+a)^(n) = P+Q" "(1)`
Similarly
`(x-a)^(n) = P - Q" "(2)`
(i) `(1) xx (2) rArr P^(2) - Q^(2) = (x^(2) - a^(2))^(n)`
(ii) Squaring (1) and (2) and subtrcting (2) form (1), we get
`4PO = (x+a)^(2n) -(x-a)^(2n)`
(iii) Squaring (1) and (2) and adding,
`2(P^(2) + Q^(2)) = (x+a)^(2n) +(x-a)^(2n)`