If `o` be the sum of odd terms and `E` that of even terms in the expansion of `(x+a)^n` prove that: (i)`O^2-E^2=(x^2-a^2)^n` (ii) `4O E=(x+a)^(2n)-(x-a)^(2n)` (iii) `2(O^2+E^2)=(x+a)^(2n)+(x-a)^(2n)`

Options

A. `(x^(2) + a^(2))^(n)`
B. `(x^(2) - a^(2))`
C. `(x-a)^(2n)`
D. none of these

Answer

b

Explanation

We have,
`(x +a)^(n) = ""^(n)C_(0) x^(n) a^(0) + ""^(n)C_(1) x^(n-1) a^(1) + ""^(n)C_(2) x^(n-2) a^(2) +... + ""^(n)C_(n-1) x a^(n-1) + ""^(n)C_(n) a^(n)`
`rArr (x +a)^(n) = ""^(n)C_(0) x^(n) a^(0) + ""^(n)C_(2) x^(n-2) a^(2) + ...) +(""^(n)C_(1) x^(n-1) a^(1) + ""^(n)C_(3)x^(n-3) a^(3)+....)`
`rArr (x + a)^(n) + O +E` ...(i)
and ,
`(x +a)^(n) = ""^(n)C_(0) a^(n) - ""^(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-1) x(-1)^(n-1) a^(n-1)+ ""^(n)C_(n) (-1)^(n) a^(n)`
`rArr (x - a)^(n) = (""^(n)C_(0) x^(n) + ""^(n)C_(2) x^(n-2) a^(2) +...)`
`- (""^(n)C_(1) x^(n-1) a^(1) + ""^(n)C_(3) x^(n-3) a^(3) +....)`
`rArr (x = a)^(n) + O - E` ...(ii)
From (i) and (ii), we get
`(x +a)^(n) (x -a)^(n) = (O +E) (O -E)`
`rArr (x^(2) - a^(2))^(n) = O^(2) - E^(2)`.