Question

If in the expansion of (x + a)ⁿ, the sum of odd terms is P and the sum of even terms is Q, then the value of (x + a)²ⁿ - (x - a)²ⁿ is:
1. P + Q
2. P - Q
3. PQ
4. 4PQ

Answer 1

Correct Answer - Option 4 : 4PQ

we have (x + a)ⁿ

\({\left( {x + a} \right)^n} = {}_{}^n{C_0}{x^n} + {}_{}^n{C_1}{x^{n - 1}}{a^1} + {}_{}^n{C_2}{x^{n - 2}}{a^2} + {}_{}^n{C_3}{x^{n - 3}}{a^3} + \ldots + {}_{}^n{C_n}{a^n}\)

Sum of odd terms,

\(P = {}_{}^n{C_0}{x^n} + + {}_{}^n{C_2}{x^{n - 2}}{a^2} + ...\)

Sum of even terms,

\(Q = {}_{}^n{C_1}{x^{n - 1}}{a^1} + {}_{}^n{C_3}{x^{n - 3}}{a^3} ...\)

⇒ (x + a)ⁿ = P + Q       (i)

⇒ (x - a)ⁿ = P - Q       (ii)

squaring and subtracting equation (ii) from equation (i) we get.

(x + a)²ⁿ - (x - a)²ⁿ = (P + Q)² - (P - Q)²

(x + a)2n - (x - a)²ⁿ = P² + Q² + 2PQ - (P² + Q² - 2PQ)

(x + a)2n - (x - a)²ⁿ = P2 + Q2 + 2PQ - P² - Q² + 2PQ

(x + a)2n - (x - a)²ⁿ = 4PQ