Question

Given the integers r > 1, n > 2, and coefficients of (3r)^th and (r + 2)^nd terms in the binomial expansion of (1 + x)²ⁿ are equal, then ______.

Options

• n = 2r

• n = 3r

• n = 2r + 1

• None of these

Answer 1

Given the integers r > 1, n > 2, and coefficients of (3r)^th and (r + 2)^nd terms in the binomial expansion of (1 + x)²ⁿ are equal, then n = 2r.

Explanation:

Given that r > 1 and n > 2

Then `"T"_(3r) = "T"_(3r - 1 + 1)`

= `""^(2n)"C"_(3r - 1) * x^(3r - 1)`

And `"T_(r + 2) = "T"_(r + 1 + 1)`

= `""^(2n)"C"_(r + 1) x^(r + 1)`

We have `""^(2n)"C"_(3r - 1) = ""^(2n)"C"_(r + 1)`

⇒ 3r – 1 + r + 1 = 2n   `....[because ""^n"C"_p = ""^n"C"_q ⇒ n = p + q]`

⇒ 4r = 2n

n = 2r