In the binomial expansion of (ax - 1 3x^2 )¹⁰ where a > 0 , the ratio of the coefficient of the 4^ th term to the coefficient of the 8^ th term is 16 . The value of 9a^2 is:

Step-by-step solution

The general term in the expansion of (x + y)^n is given by T_ r+1 = ^ n C_ r x^ n-r y^r . For the 4^ th term, r = 3 : T₄ = ¹⁰C₃ (ax)^7 (- 1 3x^2 )^3 Coefficient of 4^ th term, C₄ = ¹⁰C₃ a^7 (- 1 3 )^3 For the 8^ th term, r = 7 : T₈ = ¹⁰C₇ (ax)^3 (- 1 3x^2 )^7 Coefficient of 8^ th term, C₈ = ¹⁰C₇ a^3 (- 1 3 )^7 The ratio of the coefficients is given as 16 : C₄ C₈ = ¹⁰C₃ a^7 (- 1 3 )^3 ¹⁰C₇ a^3 (- 1 3 )^7 Since ¹⁰C₃ = ¹⁰C₇ , they cancel out: C₄ C₈ = a^7 a^3 (- 1 3 )³⁻⁷ = a^4 (- 1 3 )⁻⁴ = a^4 (-3)^4 = 81a^4 Equating t