TITLE: Constructing a recursive sequence that converges to sqrt 17
QUESTION [0 upvotes]: One of the problems that we have for abstract math is the following: Using the recursive sequence definition, construct a sequence that converges to $\sqrt{17}$.
It is my understanding that the recursive sequence definition is: $F_n=F_{n-1}+F_{n-2}$. I also found that $\sqrt{17}$ is a root of the quadratic equation: $x^2-2x-16=0$. Now I don't know how to go about constructing the sequence to converge to $\sqrt{17}$. Anyone know a good way to go about this? Any help is greatly appreciated!

REPLY [0 votes]: $$
F_n = {F_{n-1} + 17/F_{n-1} \over 2}
$$
where $F_0 > 0$, is what I use, when i need one.. it's basically the same as Steve wrote. Besides, the way I understand it a recursive definition is simply $F_n = f(F_{n-m})$ for some $m \in \mathbb{N}$, ie. basically any function that takes an old value to produce the new one.
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