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1.A Fibonacci-type sequence that has been studied by mathematicians is the Lucas sequence, named


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1.A Fibonacci-type sequence that has been studied by mathematicians is the Lucas sequence, named after a French mathematician of the nineteenth century. The first ten terms of the Lucas sequence are 1, 3, 4, 7, 11, 18, 29, 47, 76, 123. What is the eleventh term of the Lucas sequence?

The eleventh term of the Lucas sequence is

2.

The following pattern is established involving terms of the Fibonacci sequence. Use inductive reasoning to make a conjecture concerning the next equation in the pattern, and verify it. You may wish to refer to the first few terms of the sequence given in the text.

1=1

1+2=3

1+2=3

1+2+5=8

1+2+5+13=21

1+2+5+13+34=55

Each equation skips the next Fibonacci number to add the one after. Each side of the next equation will equal

Following the pattern, write the next equation.


3. What is the approximate value of the golden ratio to the nearest thousandth?

The approximate value of the golden ratio is


4.What is the exact value of the golden ratio?



5. Find the residue of 227 (mod 14)

 


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