Two lemmas on linear sequences of order two

Note: My English is so bad but don’t worry about that! 🙂

A sequence $(a_n)$ is called linear of order two if there are real numbers $p,q$ such that $a_{n+2}=pa_{n+1}+qa_n\;\forall n=1,2,3,...$

In this topic we’ll use ideas in proofs of two following lemmas to solve some Olympiad problems.

Dãy cho bởi phương trình

Trong de thi HSGQG cua Vietnam chung ta thuong gap bai toan sau: Cho cac phuong trinh $f_n(x) = 0 (n\in\mathbb{N})$ , chung minh moi phuong trinh do co nghiem thuc duy nhat tren mot mien D nao do, goi nghiem do la $x_n$, tim $\lim_{n\to\infty} x_n$.

Topic nay danh de trao doi ve nhung bai toan do, xin moi cac ban post cac loi giai va cung cap cac vi du.

Dãy dạng u_{n+1}=f(u_n)

Topic nay danh de trao doi cac bai toan gioi han cua day so kieu $u_{n+1}=f(u_n)$ (o day $f$ la mot ham so) mot kieu day xuat hien thuong xuyen trong cac bai thi HSGQG cua Vietnam chung ta. Theo toi thi chung ta chi can mot chut kien thuc ve giai tich la co the lam tot bai toan kieu nay. Toi se post bai dau tien som nhat co the. 😀