This problem investigates one connection between the Golden Ratio and the Fibonacci Sequence.
The Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, 21, . . . begins with F(1) = 1, F(2) = 1 and nth term, n > 2 is denoted by
F(n) = F(n-1) + F(n-2).
Create a Spreadsheet to generate the Fibonacci Sequence.
See the Sublime Triangle for one derivation of the Golden Ratio.
We have
and setting
the equations
can be used to find
. Now, use the first equation to generate a sequence of positive powers of the Golden Ratio:
Verify each of these by pursuing the relevant algebra, e.g.
PROVE with Mathematical Induction:
Find the sequence of powers of 
for NEGAGIVE integers.
Begin with
Continue and create a general expression to prove by mathematical induction.
