Question

coefficients of 5th, 6th and 7th terms are in A.P. for expansion of (1+x)ⁿ then n = ____

(a) 7 or 12 

(b) -7 or 14 

(c) 7 or 14 

(d) -7 or 12

Answer 1

Correct option: (c) 7 or 14

Explanation:

Coefficient of 5^th, 6^th, 7^th term in A. P.

i.e. ⁿc₄, ⁿc₅, ⁿc₆ in A. P.

∴ ⁿc₄ + ⁿc₆ = 2 ∙ ⁿc₅ 

∴ [(n!) / {4!(n – 4)!}] + [(n!) / {6!(n – 6)!}] = 2 ∙ [(n!) / {5!(n – 5)!}]

∴ [{(n – 5)!} / {(n – 4)! 4!}] + [{(n – 5)!} / {(n – 6)! 6!}] = (2 / 5!)

∴ [1 / {(n – 4)4!}] + [(n – 5) / (6!)] = (2 / 5!)

∴ [5 / (n – 4)] + [(n – 5) / 6] = 2

∴ 30 + n² – 4n – 5n + 20 = 12(n – 4)

∴ n² – 9n + 50 = 12n – 48

∴ n² – 21 + 98 = 0

n = 7, n = 14