Assignment 12

Fibonacci Sequences:

 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610

Here Is the fibonacci sequence. Notice that each term is the sum of the previous two terms.

 1 1 1 2 2 3 1.5 5 1.66666666666667 8 1.6 13 1.625 21 1.61538461538462 34 1.61904761904762 55 1.61764705882353 89 1.61818181818182 144 1.61797752808989 233 1.61805555555556 377 1.61802575107296 610 1.61803713527851

In this chart, the ratio of each two previous terms is calculated. Notice that it approaches a fixed number 1.618...This number is called phi.

 1 1 1 2 2 2 3 1.5 3 5 1.66666666666667 2.5 8 1.6 2.66666666666667 13 1.625 2.6 21 1.61538461538462 2.625 34 1.61904761904762 2.61538461538462 55 1.61764705882353 2.61904761904762 89 1.61818181818182 2.61764705882353 144 1.61797752808989 2.61818181818182 233 1.61805555555556 2.61797752808989 377 1.61802575107296 2.61805555555556 610 1.61803713527851 2.61802575107296

This chart shows the ratio of each term to the term previous to the one before it. Notice that this ratio also approaches a fixed number. In this case the number is 2.618...or phi + 1. This is the square of phi.

Here I changed the initial values. f(0) = 1 and f(1) = 3. This is called the Lucas sequence.

 1 3 3 4 1.33333333333333 4 7 1.75 2.33333333333333 11 1.57142857142857 2.75 18 1.63636363636364 2.57142857142857 29 1.61111111111111 2.63636363636364 47 1.62068965517241 2.61111111111111 76 1.61702127659574 2.62068965517241 123 1.61842105263158 2.61702127659574 199 1.61788617886179 2.61842105263158 322 1.61809045226131 2.61788617886179 521 1.61801242236025 2.61809045226131 843 1.61804222648752 2.61801242236025 1364 1.61803084223013 2.61804222648752

The first ratio still approaches phi.

Notice that the same relationship exists for the second ratio as well.

Now I am going to change f(0) = 2 and f(1) = 10. These are just arbitrary numbers that I picked.

 2 10 5 12 1.2 6 22 1.83333333333333 2.2 34 1.54545454545455 2.83333333333333 56 1.64705882352941 2.54545454545455 90 1.60714285714286 2.64705882352941 146 1.62222222222222 2.60714285714286 236 1.61643835616438 2.62222222222222 382 1.61864406779661 2.61643835616438 618 1.61780104712042 2.61864406779661 1000 1.61812297734628 2.61780104712042 1618 1.618 2.61812297734628 2618 1.61804697156984 2.618 4236 1.61802902979374 2.61804697156984

Again, the same relationship exists!

From this investigation, I conclude that regardless of the initial two numbers, the relationship will remain the same for the ratios.