The Fibonacci sequence of natural numbers, $F_0,F_1,F_2,F_3,$dots can be defined by the initial values

$F_0=0$ and $F_1=1,$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for $n2,$

List the first $10$ numbers in the Fibonacci sequence: $F_0,$dots,$F_9.$

Verify that $F_1^2+F_2^2=F_2*F_3$ and $F_1^2+F_2^2+F_3^2=F_3*F_4$

Use mathematical induction to prove AAninN,${\displaystyle \underset{k=1}{\overset{n}{}}}{F}_k^2=F_n*F_{n+1}$

Please name any people who were helpful to you, and describe how they helped.