Question

HELP!
If x_n=x_n-1+x_n-2
Then what is the limit n->infinty of x_n/x_n-1?

I know this will be (1+sqrt5)/2 but how do I prove?

Answer 1

call the limit \(x\) and solve
\[x=1+\frac{1}{x}\]

Answer 2

Satellite73 can you go back to question

Answer 3

But if I say the limit is X, what do I do to show it is equal to 1 + 1/X?

Answer 4

\[\frac{x_n}{x_{n-1}}=\frac{x_{n-1}+x_{n-2}}{x_{n-1}}\]
so
\[\frac{x_n}{x_{n-1}}=1+\frac{x_{n-1}}{x_{n-2}}\]

Answer 5

if
\[\lim_{n\to \infty}\frac{x_n}{x_{n-1}}=x\] then \[\lim_{n\to \infty}\frac{x_{n-2}}{x_{n-1}}=\frac{1}{x}\] since it is the reciprocal of the first one

Answer 6

i made a typo above sorry
it should be \[\frac{x_n}{x_{n-1}}=1+\frac{x_{n-2}}{x_{n-1}}\]

Answer 7

I understand everything but where limit x_n-2/x_n-1= 1/x?

Answer 8

Ohhhh, wait. I understand it now. So then is simplifies to X = 1+1/X and then you times both sides by X and turn it into a quadratic.