If the ratio of the 3^ rd term from the beginning to the 3^ rd term from the end in the expansion of (3x^2 - 1 x )^n is 9x^6 , then the term independent of x in the expansion is:

Step-by-step solution

The general term of the expansion (x+y)^n is T_ r+1 = ^ n C_ r x^ n-r y^r . The k^ th term from the beginning is T_k and the k^ th term from the end is T_ n-k+2 . The ratio of the k^ th term from the beginning to the k^ th term from the end is given by ( x y )^ n-2k+2 . Here, x = 3x^2 , y = - 1 x , and k = 3 . Ratio = ( 3x^2 -1/x )^ n-2(3)+2 = (-3x^3)^ n-4 . Given that the ratio is 9x^6 , we have: (-3x^3)^ n-4 = 9x^6 = (-3)^2 (x^3)^2 Comparing the exponents, n - 4 = 2 n = 6 . Now, the expansion is (3x^2 - 1 x )^6 .