If nC4, nC5, and nC6 are in A.P., then find n.
Given that nC4, nC5 and nC6 are in A.P.
We know that for three numbers a, b, c are in A.P, the following condition holds,
⇒ 2b = a + c
So,
⇒ 2nC5 = nC4 + nC6
Adding 2nC5 on both sides we get,
⇒ 4nC5 = nC4 + nC5 + nC5 + nC6
We know that nCr + nCr + 1 = n + 1Cr + 1
⇒ 4nC5 = n + 1C5 + n + 1C6
⇒ 4nC5 = n + 2C6
We know that ,
And also n! = n(n – 1)(n – 2)…………2.1
⇒
⇒
⇒
⇒ 24(n – 4) = n2 + 2n + n + 2
⇒ 24n – 96 = n2 + 3n + 2
⇒ n2 – 21n + 98 = 0
⇒ n2 – 14n – 7n + 98 = 0
⇒ n(n – 14) – 7(n – 14) = 0
⇒ (n – 7)(n – 14) = 0
⇒ n – 7 = 0 or n – 14 = 0
⇒ n = 7 or n = 14
∴ The values of n are 7 and 14.