What is the only positive Fibonacci number n which is also the nth Fibonacci number?

Question

I have NO clue on how to answer this...?

Answer 1

F0 = 0

F1 = 1

F5 = 5

since you want a prime .. . .

the number n = 5

P.S. the limiting ratio is (1 + √5)/2

Answer 2

The Fibonacci sequence is:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...

where each term is found by adding the two previous terms.  The question asks you to find an n such that the nth Fibonacci number is equal to n, and you can see from looking at the above sequence that n=5 works (n=1 also works, but that's not too interesting).  The reason that they say "its square root is related directly to the limit of the ratio of two consecutive Fibonacci numbers" is that the Fibonacci numbers are explicitly generated by the formula:

F_n = 1/sqrt(5) * [((1+sqrt(5))/2)^n - ((1-sqrt(5))/2)^n],

so the limit of the ratio of two consecutive Fibonacci numbers will turn out to be:

(1+sqrt(5))/2, which is the golden ratio.

Answer 3

1,1,2,3,5,8,...

The only possibilities are 1 and 5, but 1 is not prime.

The limit is the golden ratio = [sqrt(5) + 1]/2.