What is the first term in the Fibonacci sequence to contain 1000 digits?

Answer 1

After the first few numbers in the Fib sequence, Fib(n) is very nearly equal to (phi)n / sqrt(5) where phi is the Golden Ratio = [1+sqrt(5)]/2

[The difference is around 0.00003 by Fib(20)]

So you want the smallest n such that (phi)n / sqrt(5) ≥ 10999

Taking logs,

n*log(phi) - 0.5*log(5) ≥ 999

n*log(phi) ≥ 999 + 0.5*log(5) = 999.349

n ≥ 999.349/log(phi) = 999.349/0.2090

So n = 4781