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Project a – The decimal digits of We find the decimal digits of
First, we insert
Then, inserting
Similarly, the 4-layer fraction of the 3rd iteration is
Hence, by repeated iterations we build up the multi-layer fraction as shown
This is called a continued fraction. We now evaluate the decimal digits of Table A: Finding golden ratio by continued fractions.
Project b – Show the length A-F is Show this by applying the Pythagoras theorem to the right triangle C-D-E in figure 1. (Note that C-E = Project c – Make a logarithmic spiral pinwheel Cut out the dotted circle with a spiral pattern and mount it on the eraser end of a pencil with a thumbtack.
Project d – The ratio of two adjacent Fibonacci numbers We arrange the Fibonacci numbers in a sequence {F1, F2, F3, ...}, so that F1=1, F2=1, F3=2, …, as they are listed in the second column of the table. We compute ratio Fn / Fn-1 of the two neighboring Fibonacci numbers in the third column of the table. Can you fill in the blank entries? Compare Fn / Fn-1 with the function values of continued fraction (the second column in the table of (Project a).
Table: The ratio of two neighboring Fibonacci numbers
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by continued fractions
, which appears in the second column of the table below. Now, by dropping 1/x from fraction (2) we obtain 
and similarly 
and 
. These function values are also listed in table A. You may evaluate the remaining function values in the table. At the 9th iteration we have 
= 1.61818181... from fraction (5), agreeing with the golden ratio 
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