Question

The interval in which x(>0) must lie so that the greatest term in the expansion of (1+x)²ⁿ has the greatest coefficient is

(d) None

Answer 1

Correct option: (b) [{n / (n + 1)}, {(n + 1) / n}]

Explanation:

Given: (1 + x)²ⁿ 

Greatest term has greatest coefficient

²ⁿc_n–1 x^n–1 < ²ⁿc_n xⁿ and ²ⁿc_n+1 ∙ xⁿ⁺¹ < ²ⁿc_n ∙ xⁿ 

∴ [(²ⁿc_n–1) / (²ⁿc_n)] < x and [(²ⁿc_n) / (²ⁿc_n+1)] > x¹ 

 ∴  [{(2n)! / {(n – 1)! (n + 1)!}} / {(2n)! / {n! n!}}] < x

and

[{(2n)! / {n! n!}} / {(2n)! / {(n + 1)! (n – 1)!}}] > x

∴  [(n!)² / {(n – 1)! (n + 1)!}] < x and [{(n + 1)! (n – 1)!} / (n!)²] > x

∴  [{n(n – 1)!}² / {(n – 1)! (n + 1)!}] < x < [{(n + 1)! (n – 1)!} / {n(n – 1)!}²]

∴  [{n²(n – 1)!}^­­ / {(n + 1)!}] < x < [{(n + 1)!} / {n²(n – 1)!}^­­]

∴   [{n²(n – 1)!} / {(n + 1)(n)(n – 1)!}] < x < [{(n + 1)(n)(n – 1)!} / {n²(n – 1)!}]

∴   [n / (n + 1)] < x < [(n + 1) / n]

∴ Interval is [{n / (n + 1)}, {(n + 1) / n}]