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

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

• 2

• 7

• 11

• 14

#### Solution

If the coefficients of 2nd, 3rd and the 4th 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 + … nCnxn

Here, coefficient of 2nd term = nC1

Coefficient of 3rd term = nC2

And coefficient of 4th 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

Concept: Proof of Binomial Therom by Combination
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#### APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 8 Binomial Theorem
Exercise | Q 22 | Page 144
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