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:
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Need help finishing the proof?
Find the sequence of powers of for NEGAGIVE integers.
Continue and create a general expression to prove by mathematical induction.