What is the 1000th Fibonacci number divided by the 1001 Fibonacci number?

Answer 1

0.999

Answer 2

The 960th Fibonacci number is already larger than (1 googol)2 , so a precise,

unrounded answer to your question would have more than 10200 digits in it,

and you're not going to find anybody outside of a college supercomputer lab

who can work with the numbers you're asking about.

For the same reason, we can be pretty sure that you have no earthly use for

the answer either, so we're not particularly bothered by the fact that we can't

provide it.

However, don't despair. All is not lost! We can get you close enough for your

purpose, no matter what that purpose may be.

One nice thing about the Fibonacci series is that it's known to converge rapidly,

so we can guarantee you that, say, the 20th number divided by the 21st number,

is pretty close to the ultimate limit of the series, and darn close to the answer to

your question.

That answer is 0.618033998521803 . (rounded)

If that's not close enough for you, we have yet another way to make you happy ...

a do-it-yourself kit that you can use to find the absolute ultimate limit of the

ratio of consecutive terms of the Fibonacci series. With this, you can find it just

as accurately as you need it, just by taking a simple square root. Here it is:

One term divided by the next one is

1/2 [ sqrt(5) - 1 ]

and one term divided by the last one is

1/2 [ sqrt(5) + 1 ].

These are exact, with no messy round-offs. Each is as accurate as the

square-root in it is.

You're quite welcome.