Question

If the coefficients of 2^nd, 3^rd and the 4^th terms in the expansion of (1 + x)ⁿ are in A.P., then value of n is ______.

Options

• 2

• 7

• 11

• 14

Answer 1

If the coefficients of 2^nd, 3^rd and the 4^th terms in the expansion of (1 + x)ⁿ are in A.P., then value of n is 7.

Explanation:

Given expression is (1 + x)ⁿ

(1 + x)ⁿ = ⁿC₀ + ⁿC₁x + ⁿC₂x² + ⁿC₃x³ + … ⁿC_nxⁿ

Here, coefficient of 2^nd term = ⁿC₁

Coefficient of 3^rd term = ⁿC₂

And coefficient of 4^th term = ⁿC₃

Given that ⁿC₁, ⁿC₂ and ⁿC₃ are in A.P.

∴ 2 . ⁿC₂ = ⁿC₁ + ⁿC₃

⇒ `2 * (n(n - 1))/2 = n + (n(n - 1)(n - 2))/(3*2*1)`

⇒ `n(n - 1) = n + (n(n - 1)(n - 2))/6`

⇒ n – 1 = `1 + ((n - 1)(n - 2))/6`

⇒ 6n – 6 = 6 + n² – 3n + 2

⇒ n² – 3n – 6n + 14 = 0

⇒ n² – 9n + 14 = 0

⇒ n² – 7n – 2n + 14 = 0

⇒ n(n – 7) – 2(n – 7) = 0

⇒ (n – 2)(n – 7) = 0

⇒ n = 2, 7

⇒ n = 7

Whereas n = 2 is not possible