The coefficient of the third term in the expansion of (x^2- 1 4 )^n , when expanded in the descending power of x is 31, then n is

Step-by-step solution

The general term in the expansion of (x^2 - 1 4 )^n is given by T_ r+1 = ^ n C_ r (x^2)^ n-r (- 1 4 )^r . The third term corresponds to r = 2 . T₃ = ^ n C₂ (x^2)^ n-2 (- 1 4 )^2 = ^ n C₂ x^ 2n-4 ( 1 16 ) . The coefficient of the third term is given as 31. Since the expansion is in descending powers of x , the coefficient is independent of x if 2n-4 = 0 , but here the coefficient is simply the constant part of the term. The coefficient is ^ n C₂ 16 = 31 . Therefore, ^ n C₂ = 31 16 = 496 . Using the formula ^ n C₂ =