Establish that the formula un un1 un2 un12 1n holds for n

Establish that the formula u_n u_n-1 = u_n^2 - u_n-1^2 + (-1)^n holds for n ge 2, and use this to conclude that consecutive Fibonacci numbers are relatively prime.

Solution

For Fibonaaci Series

F0 = 0, F1 = 1, F2 = 1, F3 = 2

We can prove the given statement with the help of mathematical induction

Base Case (n=2)

F2 * F1 = F2^2 - F1^2 + (-1)^2

1 * 1 = 1^2 - 1^2 + (-1)^2

1 = 1

Hence the base case is satisfied

Assumption Step: (Let us assume that the given thing is valid for k)

uk * u(k-1) = uk^2 - u(k-1)^2 + (-1)^(k)

Induction Step

u(k+1)*u(k) = u(k+1)^2 - uk^2 + (-1)^(k+1)

The numbers will be relatively prime, since

F1 = 1, F2 = 1,F3 = 2, F4 = 3

1 and 2 are relatively prime to each other

2 and 3 are relatively prime to each other