In the number \(1234096\dots\) each digit, starting with the 5th digit is equal to the final digit of the sum of the previous 4 digits. Will the digits 8123 ever occur in that order in a row in this number?
In a row there are 2023 numbers. The first number is 1. It is known that each number, except the first and the last, is equal to the sum of two neighboring ones. Find the last number.
Prove that every pair of consecutive Fibonacci numbers are coprime. That is, they share no common factors other than 1.
Calculate the following: \(F_1^2-F_0F_2\), \(F_2^2-F_1F_3\), \(F_3^2-F_2F_4\), \(F_4^2-F_3F_5\) and \(F_5^2-F_4F_6\). What do you notice?
Work out \(F_3^2-F_0F_6\), \(F_4^2-F_1F_7\), \(F_5^2-F_2F_8\) and \(F_6^2-F_3F_9\). What pattern do you spot?
Can every whole number be written as the sum of two Fibonacci numbers? If yes, then prove it. If not, then give an example of a number that can’t be. The two Fibonacci numbers don’t have to be different.
What’s \(\sum_{i=0}^nF_i^2=F_0^2+F_1^2+F_2^2+...+F_{n-1}^2+F_n^2\) in terms of just \(F_n\) and \(F_{n+1}\)?
What are the ratios \(\frac{F_2}{F_1}\), \(\frac{F_3}{F_2}\), and so on until \(\frac{F_7}{F_6}\)? What do you notice about them?
In the example, we saw that \(\varphi^2=\varphi+1\). Can you write \(\varphi^3\) in the form \(a\varphi+b\), where \(a\) and \(b\) are positive integers?
Simplify \(F_0-F_1+F_2-F_3+...-F_{2n-1}+F_{2n}\), where \(n\) is a positive integer.