Given positive integers , and the coefficient of th and th terms in the binomial expansion of are equal. Then,
- A.
- B.
- C.
- D.None of these
Show answer and solution
Correct option: A
Solution
The coefficient of the (3r)th term in (1+x)^(2n) is C(2n, 3r-1), and the coefficient of the (r+2)th term is C(2n, r+1).
Setting them equal: C(2n, 3r-1) = C(2n, r+1).
Using the identity C(n,a) = C(n,b) implies either a = b or a + b = n:
Case 1: 3r-1 = r+1 → 2r = 2 → r = 1, but r > 1, so rejected.
Case 2: (3r-1) + (r+1) = 2n → 4r = 2n → n = 2r.
Since n > 2 and r > 1 are consistent with n = 2r, the answer is n = 2r.
Setting them equal: C(2n, 3r-1) = C(2n, r+1).
Using the identity C(n,a) = C(n,b) implies either a = b or a + b = n:
Case 1: 3r-1 = r+1 → 2r = 2 → r = 1, but r > 1, so rejected.
Case 2: (3r-1) + (r+1) = 2n → 4r = 2n → n = 2r.
Since n > 2 and r > 1 are consistent with n = 2r, the answer is n = 2r.
[1980]