The golden question

Calculus Level 2

Let $\{u_n\}$ be a sequence satisfying the recurrence relation $u_1=1$ , $u_2=2$ and $u_n=u_{n-1} + u_{n-2}$ for $n \ge 3$ .

Prove that $\displaystyle \lim_{n\to\infty} \frac{u_{n+1}}{u_n}$ exists. Calculate this limit.

\infty

\frac {1+\sqrt5}2

0 1

2 solutions

Conor Donovan
Nov 21, 2016

The recurrence relation is clearly just the Fibonacci sequence. It is well known that the ratio of consecutive Fibonacci terms tends towards the Golden Ratio which is equal to $\frac{1+\sqrt{5}}{2}$

Can you show ts proof?

Puneet Pinku - 4 years, 5 months ago

Irina Stanciu
Nov 19, 2016

We can deduce that u(n+1)* u(n+1)/ u(n) * u(n) - u(n)( u (n) + u (n+1)/ u(n)* u(n)= (-1) pow (n+1) / u(n) * u(n). The succesive values for u(n) are: 1,2,3,5,8,13,...,u(n) -> infinte with n.
( u (n+1)/u(n) ) pow 2 - (u (n+1) / u(n)) -1= (-1) pow (n+1) / u (n) pow 2.
We say that when n-> infinte X= lim (u (n+1)/ u (n) ) X X - X -1=0 => X= ( 1 + sqrt5)/2=1,618033989...,
for a=1 we got b
b -b -1=0 => b=X.

It's hardly understandable without Latex :(

Peter van der Linden - 4 years, 6 months ago

I will rewrite it on a piece of paper and post it! :D

Irina Stanciu - 4 years, 6 months ago

Umm...it's been years and u hvnt posted😅

Rahul Gautam - 3 years, 3 months ago

The golden question

2 solutions

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