# Assignment 12.4

For this assignment, I have chose to investigate the Fibbonnaci sequence.  I had never heard of the Fibbonnaci sequence until my third year of college, so there is a lot I do not know.   At this point all I do know is that it is generated by f(n) = f(n-1) + f(n-2).  I will begin by generating the Fibbonnaci sequence on a spreadsheet.  I will generate it up to f(25) and such that f(0) = 1 and f(1)= 1.

 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025

Now that we can see how the Fibbonnaci sequence is constructed and what it looks like, we can begin to investigate it.  I will start by taking the ratio of the adjacent terms or more formally g(n) = f(n-1)/f(n-2).

 1 1 1 2 2 3 1.5 5 1.666666667 8 1.6 13 1.625 21 1.615384615 34 1.619047619 55 1.617647059 89 1.618181818 144 1.617977528 233 1.618055556 377 1.618025751 610 1.618037135 987 1.618032787 1597 1.618034448 2584 1.618033813 4181 1.618034056 6765 1.618033963 10946 1.618033999 17711 1.618033985 28657 1.61803399 46368 1.618033988 75025 1.618033989

As we can see, as f(n) gets larger g(n) approaches 1.618, which is the golden ratio.  This was not too surprising to me because I had seen it before, I just did not remember it.  Next, I start playing around with the f(0) and f(1) terms.  I changed them to one and three, as the assignment recommended.  I then took the ratio of its adjacent terms.

 1 3 3 4 1.333333 7 1.75 11 1.571429 18 1.636364 29 1.611111 47 1.62069 76 1.617021 123 1.618421 199 1.617886 322 1.61809 521 1.618012 843 1.618042 1364 1.618031 2207 1.618035 3571 1.618034 5778 1.618034 9349 1.618034 15127 1.618034 24476 1.618034 39603 1.618034 64079 1.618034 103682 1.618034 167761 1.618034

This surprised me somewhat, but not too much because I figured it was a special case.  I then began to plug in various numbers for f(0) and f(1).  To my surprise, the golden ratio came out every time.  Here are a couple of examples…

F(0) = 1 and f(1) = 3

 1 3 3 4 1.333333 7 1.75 11 1.571429 18 1.636364 29 1.611111 47 1.62069 76 1.617021 123 1.618421 199 1.617886 322 1.61809 521 1.618012 843 1.618042 1364 1.618031 2207 1.618035 3571 1.618034 5778 1.618034 9349 1.618034 15127 1.618034 24476 1.618034 39603 1.618034 64079 1.618034 103682 1.618034 167761 1.618034

F(0) = 0 and f(1) = 2

 0 2 2 1 4 2 6 1.5 10 1.666667 16 1.6 26 1.625 42 1.615385 68 1.619048 110 1.617647 178 1.618182 288 1.617978 466 1.618056 754 1.618026 1220 1.618037 1974 1.618033 3194 1.618034 5168 1.618034 8362 1.618034 13530 1.618034 21892 1.618034 35422 1.618034 57314 1.618034 92736 1.618034

F(0) = 300 and f(1) = 1

 300 1 0.003333 301 301 302 1.003322 603 1.996689 905 1.500829 1508 1.666298 2413 1.600133 3921 1.624948 6334 1.615404 10255 1.61904 16589 1.61765 26844 1.618181 43433 1.617978 70277 1.618055 113710 1.618026 183987 1.618037 297697 1.618033 481684 1.618034 779381 1.618034 1261065 1.618034 2040446 1.618034 3301511 1.618034 5341957 1.618034 8643468 1.618034

F(0) = -6 and f(1) = -2

 -6 -2 0.333333 -8 4 -10 1.25 -18 1.8 -28 1.555556 -46 1.642857 -74 1.608696 -120 1.621622 -194 1.616667 -314 1.618557 -508 1.617834 -822 1.61811 -1330 1.618005 -2152 1.618045 -3482 1.61803 -5634 1.618036 -9116 1.618033 -14750 1.618034 -23866 1.618034 -38616 1.618034 -62482 1.618034 -101098 1.618034 -163580 1.618034 -264678 1.618034

Now I will try taking the ratio of every other term or h(n) = f(n-1)/f(n-3).  I have no idea what the ratio will be.

F(0) = 1 and f(1) = 1

 1 1 2 2 3 3 5 2.5 8 2.666667 13 2.6 21 2.625 34 2.615385 55 2.619048 89 2.617647 144 2.618182 233 2.617978 377 2.618056 610 2.618026 987 2.618037 1597 2.618033 2584 2.618034 4181 2.618034 6765 2.618034 10946 2.618034 17711 2.618034 28657 2.618034 46368 2.618034 75025 2.618034

As you can see, the ratio seems to be the golden ratio plus one.  Now I am going to check to see if this works for other values of f(0) and f(1).

F(0) = 0 and f(1) = 2

 0 2 2 #DIV/0! 4 2 6 3 10 2.5 16 2.666667 26 2.6 42 2.625 68 2.615385 110 2.619048 178 2.617647 288 2.618182 466 2.617978 754 2.618056 1220 2.618026 1974 2.618037 3194 2.618033 5168 2.618034 8362 2.618034 13530 2.618034 21892 2.618034 35422 2.618034 57314 2.618034 92736 2.618034

F(0) = -7 and f(1) = -99

 -7 -99 -106 15.14286 -205 2.070707 -311 2.933962 -516 2.517073 -827 2.659164 -1343 2.602713 -2170 2.623942 -3513 2.615786 -5683 2.618894 -9196 2.617706 -14879 2.618159 -24075 2.617986 -38954 2.618052 -63029 2.618027 -101983 2.618037 -165012 2.618033 -266995 2.618034 -432007 2.618034 -699002 2.618034 -1131009 2.618034 -1830011 2.618034 -2961020 2.618034 -4791031 2.618034

F(0) = .001 and f(1) = .35

 0.35 0.001 0.351 1.002857 0.352 352 0.703 2.002849 1.055 2.997159 1.758 2.500711 2.813 2.666351 4.571 2.600114 7.384 2.624956 11.955 2.615401 19.339 2.619041 31.294 2.61765 50.633 2.618181 81.927 2.617978 132.56 2.618055 214.487 2.618026 347.047 2.618037 561.534 2.618033 908.581 2.618034 1470.12 2.618034 2378.7 2.618034 3848.81 2.618034 6227.51 2.618034 10076.3 2.618034

F(0) = -654 and f(1) = 1.2

 -654 1.2 -652.8 0.998165 -651.6 -543 -1304.4 1.998162 -1956 3.001842 -3260.4 2.49954 -5216.4 2.666871 -8476.8 2.599926 -13693.2 2.625029 -22170 2.615374 -35863.2 2.619052 -58033.2 2.617645 -93896.4 2.618182 -151930 2.617977 -245826 2.618056 -397756 2.618026 -643582 2.618037 -1.04134e+06 2.618033 -1.68492e+06 2.618034 -2.72626e+06 2.618034 -4.41118e+06 2.618034 -7.13743e+06 2.618034 -1.2e+07 2.618034 -1.9e+07 2.618034

I tried to pick the most random values for f(0) and f(1) and no matter what their value was the ratio was always the gold ratio plus one.

My final conclusions for this assignment are…

F(n) = f(n-1) + f(n-2)  => f(n-1)/f(n-2) = 1.61… = (1 + sqrt(5)) / 2 = the golden ratio.

F(n) = f(n-1) + f(n-2)  => f(n-1)/f(n-3) = 2.61… =[(1 + sqrt(5)) / 2] + 1 = the golden ratio + 1.