Rayen Antillanca. Assignment 12

Generate a Fibonnaci sequence in the first column using f(0) = 1, f(1) = 1,f(n) = f(n-1) + f(n-2)

 Fibonnaci Sequence n Fibonnaci Sequence 1 1 2 1 1 3 2 2 2 4 3 1.5 3 3 5 5 1.666667 2.5 5 5 6 8 1.6 2.666667 4 8 7 13 1.625 2.6 4.333333 6.5 8 21 1.615385 2.625 4.2 7 9 34 1.619048 2.615385 4.25 6.8 10 55 1.617647 2.619048 4.230769 6.875 11 89 1.618182 2.617647 4.238095 6.846154 12 144 1.617978 2.618182 4.235294 6.857143 13 233 1.618056 2.617978 4.236364 6.852941 14 377 1.618026 2.618056 4.235955 6.854545 15 610 1.618037 2.618026 4.236111 6.853933 16 987 1.618033 2.618037 4.236052 6.854167 17 1597 1.618034 2.618033 4.236074 6.854077 18 2584 1.618034 2.618034 4.236066 6.854111 19 4181 1.618034 2.618034 4.236069 6.854098 20 6765 1.618034 2.618034 4.236068 6.854103 21 10946 1.618034 2.618034 4.236068 6.854101 22 17711 1.618034 2.618034 4.236068 6.854102 23 28657 1.618034 2.618034 4.236068 6.854102 24 46368 1.618034 2.618034 4.236068 6.854102 25 75025 1.618034 2.618034 4.236068 6.854102 26 121393 1.618034 2.618034 4.236068 6.854102 27 196418 1.618034 2.618034 4.236068 6.854102 28 317811 1.618034 2.618034 4.236068 6.854102 29 514229 1.618034 2.618034 4.236068 6.854102 30 832040 1.618034 2.618034 4.236068 6.854102 31 1346269 1.618034 2.618034 4.236068 6.854102 32 2178309 1.618034 2.618034 4.236068 6.854102 33 3524578 1.618034 2.618034 4.236068 6.854102 34 5702887 1.618034 2.618034 4.236068 6.854102 35 9227465 1.618034 2.618034 4.236068 6.854102

The next graphs show what happens when n increases.

 When n increases, the first ratio, the ratio between each adjacent pair of numbers in the Fibonnaci Sequence, is approaching to 1.618033089 which is the Golden Ratio When n increases, the first ratio, the ratio between the n-th and the  (n-2)-th terms of the Fibonnaci Sequence, is approaching to 2.618033989 which is the square of Golden Ratio When n increases, the first ratio, the ratio between the n-th and the  (n-3)-th terms of the Fibonnaci Sequence, is approaching 4.236067977 which is the cubic of Golden Ratio. When n increases, the first ratio, the ratio between the n-th and the  (n-4)-th terms of the Fibonnaci Sequence, is approaching 6.854101966 Therefore, as n increases, we observe that the limits of the ratios above explored correspond to the Golden Ratio, Square of Golden Ratio, Cubic Golden Ratio, 4th power of Golden Ratio, etc. respectively.

 Lucas Sequence n Lucas Sequence 1 3 3 2 4 1.333333 4 3 7 1.75 2.333333 7 4 11 1.571429 2.75 3.666667 11 5 18 1.636364 2.571429 4.5 6 6 29 1.611111 2.636364 4.142857 7.25 7 47 1.62069 2.611111 4.272727 6.714286 8 76 1.617021 2.62069 4.222222 6.909091 9 123 1.618421 2.617021 4.241379 6.833333 10 199 1.617886 2.618421 4.234043 6.862069 11 322 1.61809 2.617886 4.236842 6.851064 12 521 1.618012 2.61809 4.235772 6.855263 13 843 1.618042 2.618012 4.236181 6.853659 14 1364 1.618031 2.618042 4.236025 6.854271 15 2207 1.618035 2.618031 4.236084 6.854037 16 3571 1.618034 2.618035 4.236062 6.854127 17 5778 1.618034 2.618034 4.23607 6.854093 18 9349 1.618034 2.618034 4.236067 6.854106 19 15127 1.618034 2.618034 4.236068 6.854101 20 24476 1.618034 2.618034 4.236068 6.854102 21 39603 1.618034 2.618034 4.236068 6.854102 22 64079 1.618034 2.618034 4.236068 6.854102 23 103682 1.618034 2.618034 4.236068 6.854102 24 167761 1.618034 2.618034 4.236068 6.854102 25 271443 1.618034 2.618034 4.236068 6.854102 26 439204 1.618034 2.618034 4.236068 6.854102 27 710647 1.618034 2.618034 4.236068 6.854102 28 1149851 1.618034 2.618034 4.236068 6.854102 29 1860498 1.618034 2.618034 4.236068 6.854102 30 3010349 1.618034 2.618034 4.236068 6.854102 31 4870847 1.618034 2.618034 4.236068 6.854102 32 7881196 1.618034 2.618034 4.236068 6.854102 33 12752043 1.618034 2.618034 4.236068 6.854102 34 20633239 1.618034 2.618034 4.236068 6.854102 35 33385282 1.618034 2.618034 4.236068 6.854102

Even though in the Lucas Sequence the ratios start at different numbers, as n increase the ratios approach to the same numbers of Fibonnaci Sequence: Golden Ratio, square of Golden Ratio, Cubic of Golden Ratio, 4th power of golden Ratio respectively.

Others sequences:

 Sequence with  and n Sequence 1 5 1.25 2 9 1.8 2.25 3 14 1.555556 2.8 3.5 4 23 1.642857 2.555556 4.6 5.75 5 37 1.608696 2.642857 4.111111 7.4 6 60 1.621622 2.608696 4.285714 6.666667 7 97 1.616667 2.621622 4.217391 6.928571 8 157 1.618557 2.616667 4.243243 6.826087 9 254 1.617834 2.618557 4.233333 6.864865 10 411 1.61811 2.617834 4.237113 6.85 11 665 1.618005 2.61811 4.235669 6.85567 12 1076 1.618045 2.618005 4.23622 6.853503 13 1741 1.61803 2.618045 4.23601 6.854331 14 2817 1.618036 2.61803 4.23609 6.854015 15 4558 1.618033 2.618036 4.236059 6.854135 16 7375 1.618034 2.618033 4.236071 6.854089 17 11933 1.618034 2.618034 4.236067 6.854107 18 19308 1.618034 2.618034 4.236068 6.8541 19 31241 1.618034 2.618034 4.236068 6.854103 20 50549 1.618034 2.618034 4.236068 6.854102 21 81790 1.618034 2.618034 4.236068 6.854102 22 132339 1.618034 2.618034 4.236068 6.854102 23 214129 1.618034 2.618034 4.236068 6.854102 24 346468 1.618034 2.618034 4.236068 6.854102 25 560597 1.618034 2.618034 4.236068 6.854102 26 907065 1.618034 2.618034 4.236068 6.854102 27 1467662 1.618034 2.618034 4.236068 6.854102 28 2374727 1.618034 2.618034 4.236068 6.854102 29 3842389 1.618034 2.618034 4.236068 6.854102 30 6217116 1.618034 2.618034 4.236068 6.854102 31 10059505 1.618034 2.618034 4.236068 6.854102 32 16276621 1.618034 2.618034 4.236068 6.854102 33 26336126 1.618034 2.618034 4.236068 6.854102 34 42612747 1.618034 2.618034 4.236068 6.854102 35 68948873 1.618034 2.618034 4.236068 6.854102

 Sequence with and n 1 1 7 7 2 8 1.142857 8 3 15 1.875 2.142857 15 4 23 1.533333 2.875 3.285714 23 5 38 1.652174 2.533333 4.75 5.428571 6 61 1.605263 2.652174 4.066667 7.625 7 99 1.622951 2.605263 4.304348 6.6 8 160 1.616162 2.622951 4.210526 6.956522 9 259 1.61875 2.616162 4.245902 6.815789 10 419 1.617761 2.61875 4.232323 6.868852 11 678 1.618138 2.617761 4.2375 6.848485 12 1097 1.617994 2.618138 4.235521 6.85625 13 1775 1.618049 2.617994 4.236277 6.853282 14 2872 1.618028 2.618049 4.235988 6.854415 15 4647 1.618036 2.618028 4.236098 6.853982 16 7519 1.618033 2.618036 4.236056 6.854148 17 12166 1.618034 2.618033 4.236072 6.854085 18 19685 1.618034 2.618034 4.236066 6.854109 19 31851 1.618034 2.618034 4.236069 6.854099 20 51536 1.618034 2.618034 4.236068 6.854103 21 83387 1.618034 2.618034 4.236068 6.854102 22 134923 1.618034 2.618034 4.236068 6.854102 23 218310 1.618034 2.618034 4.236068 6.854102 24 353233 1.618034 2.618034 4.236068 6.854102 25 571543 1.618034 2.618034 4.236068 6.854102 26 924776 1.618034 2.618034 4.236068 6.854102 27 1496319 1.618034 2.618034 4.236068 6.854102 28 2421095 1.618034 2.618034 4.236068 6.854102 29 3917414 1.618034 2.618034 4.236068 6.854102 30 6338509 1.618034 2.618034 4.236068 6.854102 31 10255923 1.618034 2.618034 4.236068 6.854102 32 16594432 1.618034 2.618034 4.236068 6.854102 33 26850355 1.618034 2.618034 4.236068 6.854102 34 43444787 1.618034 2.618034 4.236068 6.854102 35 70295142 1.618034 2.618034 4.236068 6.854102

 Sequence with and n Sequence 1 3 -3 2 2 0.666667 -2 3 5 2.5 1.666667 -5 4 7 1.4 3.5 2.333333 -7 5 12 1.714286 2.4 6 4 6 19 1.583333 2.714286 3.8 9.5 7 31 1.631579 2.583333 4.428571 6.2 8 50