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

#### Solution

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} = ^{n}C_{0} + ^{n}C_{1}x + ^{n}C_{2}x^{2} + ^{n}C_{3}x^{3} + … ^{n}C_{n}x^{n}

Here, coefficient of 2^{nd} term = ^{n}C_{1}

Coefficient of 3^{rd} term = ^{n}C_{2}

And coefficient of 4^{th} term = ^{n}C_{3}

Given that ^{n}C_{1}, ^{n}C_{2} and ^{n}C_{3} are in A.P.

∴ 2 . ^{n}C_{2} = ^{n}C_{1} + ^{n}C_{3}

⇒ `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 + n^{2} – 3n + 2

⇒ n^{2} – 3n – 6n + 14 = 0

⇒ n^{2} – 9n + 14 = 0

⇒ n^{2} – 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