Consider the binomial expansion of (ax + 1 bx^2 )^n , where a, b > 0 . If the 5^ th term is independent of x , and the ratio of the coefficient of the 5^ th term to the coefficient of the 9^ th term is 81 , then the value of the product ab is:

Step-by-step solution

The general term in the expansion is T_ r+1 = ^ n C_ r (ax)^ n-r ( 1 bx^2 )^r = ^ n C_ r a^ n-r b^ -r x^ n-3r . For the 5^ th term, r = 4 . Since it is independent of x , the power of x must be zero: n - 3(4) = 0 n = 12 . Now, we find the coefficients of the 5^ th and 9^ th terms. Coefficient of the 5^ th term ( r=4 ): C₅ = ¹²C₄ a¹²⁻⁴ b⁻⁴ = ¹²C₄ a^8 b⁻⁴ Coefficient of the 9^ th term ( r=8 ): C₉ = ¹²C₈ a¹²⁻⁸ b⁻⁸ = ¹²C₈ a^4 b⁻⁸ The ratio of these coefficients is given as 81 : C₅ C₉ = ¹²C₄ a^8 b⁻⁴ ¹²C₈ a^4 b⁻⁸ Since ¹