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. none of these

Explanation

`because (x + a)^(n) = ""^(n)C_(0) x^(n-0) x^(n-1) a^(1) + ""^(n)C_(2) x^(n-2) a^(2) `
` + ""^(n)C_(3) x^(n-3) a^(3) + ... + ""^(n)C_(n) x^(n-n) a^(n)`
` = (""^(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^(1) + ""^(n)C_(3) x^(n-3) a^(3) + ""^(n)C_(5) x^(n-5) a^(5) + ...)`
= P + Q (given) ...(i)
and `(x- a)^(n) = ""^(n)C_(0) x^(n-0) a^(0) - ""^(n)C_(1) x^(n-1) a^(1) + ""^(n)C_(2) x^(n-2) a^(2)`
` - ""^(n)C_(3) x^(n-3) a^(3) + ...+ ""^(n)C_(n) x^(n-n) a^(n)`
` = ( ""^(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) + ...)`
P - Q (given) ...(ii)
(i) ` P^(2) - Q^(2) = (P + Q)(p -Q)`
` = (x + a)^(n) . (x -a)^(n)`
` = (x^(2) - a^(2))^(n) " " ` [ from Eqs (i) and (ii)]
(ii) ` (x + a)^(2n) - (x - a)^(2n) = [(x + a)^(n)]^(2) - [(x - a)^(n)]^(2)`
` = (P + Q)^(2) - (P- Q)^(2)`
` = 4PQ " "` [ from Eqs . (i) and (ii)]