Fibonacci Sequence Related Calculations

Question

Can we find y₁₀₀ of the Fibonacci sequence y_n+2 = y_n+1 + y_n with y₀ = 1, y₁ = 1 without writing the full series?

If so, how can we do that?

Answer 1

F_n = ( φⁿ - ψⁿ ) / √(5)

φ = ( 1 + √(5) ) / 2 ≈ 1.618...

ψ = 1 - φ = ( 1 - √(5) ) / 2 ≈ 0.618

n = 100: F₁₀₀ = ( φ¹⁰⁰ - ψ¹⁰⁰ ) / √(5)

∴ F₁₀₀ ≈ 354224848179261915075 ≈ 3.542 ( 10²⁰ )

Note: I don't think (this could be wrong) any normal calculator can handle that high of power/number of digits. Online calculators were used in the evaluation of the above number.

I hope this helps! Message me in the comments with any questions, comments, or concerns!

Answer 2

1,1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377. 610, 987, 1597, y18, y19, y20, ... yn, yn+1, ... , y99, y100, ...

yn = (1/sqr5)[ (1+sqr5)/2)^(n) + (1-sqr5)/2)^(n)]

y100 = (1/sqr5)[ ((1+sqr5)/2)^(100) + ((1-sqr5)/2)^100))]

y100 = 354,224, 848, 179,261, 915, 075

= about 354 quintillion= about 3.5 x 10^20