Question

[Pham Trong Bach]

Đặt f n = n 2 + n + 1 2 + 1.  Xét dãy số u n  sao cho  u n = f 1 . f 3 . f 5 ... f 2 n − 1 f 2 . f 4 . f 6 ... f 2 n . lim   n u n .

A. lim   n u n = 2

B. lim   n u n = 1 3

C. lim   n u n = 3

D. lim   n u n 1 2

Answer 1

[Cao Minh Tâm]

Đáp án là D.

Ta có

f n = n 2 + 1 + n 2 + 1 = n 2 + 1 2 + 2 n . n 2 + 1 + n 2 + 1 = n 2 + 1 n 2 + 1 + 2 n + 1

= n 2 + 1 n + 1 2 + 1

Do đó: f 2 n − 1 f 2 n = 2 n − 1 2 + 1 2 n 2 + 1 2 n 2 + 1 2 n + 1 2 + 1 = 2 n − 1 2 + 1 2 n + 1 2 + 1

Suy ra

  u n = f 1 . f 3 . f 5 ... f 2 n − 1 f 2 . f 4 . f 6 ... f 2 n = f 1 f 2 ⋅ f 3 f 4 ⋅ f 5 f 6 ⋅ ⋅ ⋅ f 2 n − 1 f 2 n

= 1 2 + 1 3 2 + 1 ⋅ 3 2 + 1 5 2 + 1 ⋅ 5 2 + 1 7 2 + 1 ⋅ ⋅ ⋅ 2 n − 1 2 + 1 2 n + 1 2 + 1 = 2 2 n + 1 2 + 1 = 1 2 n 2 + 2 n + 1

⇒ n u n = n . 1 2 n 2 + 2 n + 1

⇒ lim   n u n = 1 2