Question

If in the expansion of (1 + x)²ⁿ, coefficient of 3^rd and (r + 2)^th term are equal, then :

(A) n = 2r

(B) n = 2r – 1

(C) n = 2r + 1

(D) n = r + 1

Answer 1

Answer is (A) n = 2r

In the expansion of (1 + x)²ⁿ, 3^rd term = 2ⁿC_{3r – 1} and r + 2^th term = 2ⁿC_{r + 1}

According to question,

2ⁿC_{3r – 1} = 2ⁿC_{r + 1}

Now, 3r – 1 = r + 1 or 3r – 1 = 2n – r – 1

⇒ 3r – r – I + 1 or 3r + r = 2n – 1 + 1

⇒ 2r = 2 or 4r = 2n

⇒ r =1 or 2r = n