Question

If `o` be the sum of odd terms and `E` that of even terms in the expansion of `(x+a)^n` prove that: `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)`

Answer 1

(i) Given expansin is `(x + a)^(n)`
`:. (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_(3) x^(n - 3) a^(3) + .... + .^(n)C_(n) a^(n)`
Now, sum of odd terms
i.e., `O = .^(n)C_(0) x^(n) + .^(n)C_(2) x^(n - 2) a^(2) +....`
i.e., `E = .^(n)C_(1) x^(n - 1) a + .^(n)C_(3) x^(n - 3) a^(3) +`.....
`:. (x + a)^(n) = O + E`....(i)
Similarly `(x - a)^(n) = O - E` .....(ii)
`.: (O + E) (O - E) = (x + a)^(n) (x - a)^(n)` [on multiplying Eqs. (i) and (ii)]
`rArr O^(2) - E^(2) = (x^(2) - a^(2))^(n)`
(ii) `4 OE = (O + E)^(2) - (O - E)^(2) = [(x + a)^(n)]^(2) - [(x - a)^(n)]^(2)` [from Eqs. (i) and (ii)]
`= (x + a)^(2n) - (x - a)^(2n)`