$\{a_n\}$ is a real sequence, $a_1 > 0 $, $a_2 > 0$ and for all $n>2$ : $$a_{n+2} =\frac{2}{a_n + a_{n+1}}.$$
Prove that: $\lim_{n\to \infty} a_n = 1$.
Asked
Active
Viewed 77 times
0
-
2Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Also, many find the use of imperative ("Prove", "Solve", etc.) to be rude when asking for help; please consider rewriting your post. – Clement C. Oct 21 '15 at 15:04
-
why don't you prove it yourself? :-) – Math-fun Oct 21 '15 at 15:09
1 Answers
-1
Think of the recursion as taking a harmonic mean. Then you keep getting values that are closer and closer together, and when they are very close the answer is very close to $1$.
user2566092
- 26,142
-
2That's really far from a proof. With your argument, the recursion $a_{n+1}=1/a_n$ would be convergent. – Martin Argerami Oct 21 '15 at 15:27