Let E be the sum of the coefficients of even powers of x and O be the sum of the coefficients of odd powers of x in the expansion of (1 - 3x + 5x^2)^n . The value of E^2 - O^2 is :

Step-by-step solution

Let P(x) = (1 - 3x + 5x^2)^n . The sum of all coefficients is obtained by substituting x = 1 : P(1) = (1 - 3(1) + 5(1)^2)^n = 3^n Substituting x = -1 gives: P(-1) = (1 - 3(-1) + 5(-1)^2)^n = (1 + 3 + 5)^n = 9^n The sum of the coefficients of even powers of x is E = P(1) + P(-1) 2 . The sum of the coefficients of odd powers of x is O = P(1) - P(-1) 2 . The required expression is E^2 - O^2 . Using the difference of squares identity: E^2 - O^2 = (E + O)(E - O) Substituting the expressions for E and O : E + O = P(1) E