In the binomial expansion of (x + x^2 )^n , the term independent of x is the 4^ th term from the beginning, and its value is 672 . If x = 2 , then the ratio of the 4^ th term from the beginning to the 4^ th term from the end is:

Step-by-step solution

The general term in the expansion of (x + x^2 )^n is T_ r+1 = ^ n C_ r x^ n-r ( x^2 )^r = ^ n C_ r ^r x^ n-3r . The 4^ th term from the beginning corresponds to r = 3 . Since the 4^ th term is independent of x , the exponent of x must be zero: n - 3(3) = 0 n = 9 . The value of this term is given as 672 . T₄ = ⁹C₃ ^3 = 672 9 8 7 3 2 1 ^3 = 672 84 ^3 = 672 ^3 = 8 = 2 . The ratio of the k^ th term from the beginning to the k^ th term from the end in (a+b)^n is ( a b )^ n-2k+2 . For k = 4 , a = x , and b = x^2 , the ra