If the coefficients of the 2nd, 3rd, and 4th terms in the expansion of (1+x)^n are in arithmetic progression, then the term independent of x in the expansion of (x + x^2 - 3x^5 ) (x - 2 x^2 )^n is

Step-by-step solution

The coefficients of the 2nd, 3rd, and 4th terms in the expansion of (1+x)^n are ^ n C₁ , ^ n C₂ , and ^ n C₃ respectively. Since they are in arithmetic progression: 2 ^ n C₂ = ^ n C₁ + ^ n C₃ 2 ( n(n-1) 2 ) = n + n(n-1)(n-2) 6 Since n 3 (as the 4th term exists), we can divide by n : n - 1 = 1 + (n-1)(n-2) 6 n - 2 = n^2 - 3n + 2 6 6n - 12 = n^2 - 3n + 2 n^2 - 9n + 14 = 0 (n-2)(n-7) = 0 Since n 3 , we must have n = 7 . Now, we need the term independent of x in (x + x^2 - 3x^5 ) (x - 2 x^2 )^7 . The general term of (x