Define F(n) as the number of ways to express a natural number n as a sum of 1's and 2's. Order matters.
Example F(4)=5 from 4 = 1+1+1+1 = 2+1+1 = 1+2+1 = 1+1+2 = 2+2.

Define G(n) as the number of ways to express a natural number n as a sum of odd natural numbers. Order matters.
Example G(4)=3 from 4 = 1+1+1+1 = 1+3 = 3+1

Evaluate the limit of F(n)/G(n) as n goes to infinity.

It is a fun thing to prove with middle schoolers but both sequences lead to the Fibonacci numbers f(n), but offset by 1 term.

Posted by Jer on 2023-11-16 08:50:35