In the binomial expansion of (x^2 - 2 x^3 )^n , the ratio of the k^ th term from the end to the k^ th term from the beginning is given as -8x⁻¹⁵ . If the k^ th term from the beginning is independent of x , then the value of n + k is

Step-by-step solution

Let the binomial be (a+b)^n where a = x^2 and b = -2x⁻³ . The k^ th term from the end is T_ n-k+2 and the k^ th term from the beginning is T_k . The ratio of the k^ th term from the end to the k^ th term from the beginning is: T_ n-k+2 T_k = ^nC_ n-k+1 a^ k-1 b^ n-k+1 ^nC_ k-1 a^ n-k+1 b^ k-1 Since ^nC_ n-k+1 = ^nC_ k-1 , this simplifies to: ( b a )^ n-2k+2 = ( -2x⁻³ x^2 )^ n-2k+2 = (-2x⁻⁵)^ n-2k+2 We are given that this ratio is -8x⁻¹⁵ , which can be written as (-2)^3 x⁻¹⁵ . Comparing the two expressions, we get: