Question

If A and B are coefficient of x n in the expansions of (1 + x)²ⁿ and (1 + x)^2n–1 respectively, then `A/B` equals ______.

Options

• 1

• 2

• `1/2`

• `1/"n"`

Answer 1

If A and B are coefficient of x n in the expansions of (1 + x)²ⁿ and (1 + x)^2n–1 respectively, then `A/B` equals 2.

Explanation:

Given expression is (1 + x)²ⁿ

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

∴ Coefficient of xⁿ = ²ⁿC_n = A  .....(Given)

In the expression (1 + x)^2n–1

`"T"_(r + 1) = ""^(2n - 1)"C"_rx^r`

∴ Coefficient of xⁿ = `""^(2n - 1)"C"_n` = B  ....(Given)

So, `A/B = (""^(2n)"C"_n)/(""^(2n - 1)"C"_n)`

= `(""^(2n)"C"_n)/(""^(2n - 1)"C"_n) = 2/1`  ....[From Q. no 21]