\S 8 Some properties of Fibonacci numbers
It will be convenient to consider here the Fibonacci series defined by the conditions F_{n+1}=F_{n}+F_{n-1}, F_0=0, F_1=1.
Table 3
n F_n n F_n n F_n
-3 2 4 3 11 89
-2 -1 5 5 12 144
-1 1 6 8 13 233
0 0 7 13 14 377
1 1 8 21 15 610
2 1 9 34 16 987
3 2 10 55 17 1597
It will be seen from inspection of Table 3 that the large Fibonacci numbers are very nearly in geometric progression. This suggests putting $X_n=u^n$ in the recurrence relation $X_{n+1}=X_{n}+X_{n-1}$.