Fibonacci sequence

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# Princess Browne

Generate a Fibonacci sequence in the first column using f (0) = 1, f (1) = 1,

F (n) = f (n-1) + f (n-2)

a.  Construct the ratio of each pair of adjacent terms in the Fibonacci sequence. What happens as n increases? What about the ratio of every second term? etc.

b.   Explore sequences where f (0) and f (1) are some arbitrary integers other than 1. If f (0) =1 and f (1) = 3, then your sequence is a Lucas Sequence. All such sequences, however, have the same limit of the ratio of successive terms.

The Fibonacci numbers are a sequence of numbers named after Leonardo of Pisa, who was also known as Fibonacci.

The following relation defines the Fibonacci sequence:

The following spreadsheet will use the formula of f (n) = f (n-1) + f (n -2) to show the value of the Golden Ratio.

Formula:  Xn = F (n=1)/ F (n) = F (n) + F (n-1)/ F (n) = 1 + F (n-1)/F (n) = 1 + 1/ (F (n)/F (n-1)) = 1 + 1/Xn - 1

 1 1 1 2 2 2 3 1.5 3 3 5 1.666666667 2.5 5 5 8 1.6 2.666666667 4 8 13 1.625 2.6 4.333333333 6.5 21 1.615384615 2.625 4.2 7 34 1.619047619 2.615384615 4.25 6.8 55 1.617647059 2.619047619 4.230769231 6.875 89 1.618181818 2.617647059 4.238095238 6.846153846 144 1.617977528 2.618181818 4.235294118 6.857142857 233 1.618055556 2.617977528 4.236363636 6.852941176 377 1.618025751 2.618055556 4.235955056 6.854545455 610 1.618037135 2.618025751 4.236111111 6.853932584 987 1.618032787 2.618037135 4.236051502 6.854166667 1597 1.618034448 2.618032787 4.236074271 6.854077253 2584 1.618033813 2.618034448 4.236065574 6.854111406 4181 1.618034056 2.618033813 4.236068896 6.854098361 6765 1.618033963 2.618034056 4.236067627 6.854103343 10946 1.618033999 2.618033963 4.236068111 6.85410144 17711 1.618033985 2.618033999 4.236067926 6.854102167 28657 1.61803399 2.618033985 4.236067997 6.85410189 46368 1.618033988 2.61803399 4.23606797 6.854101996 75025 1.618033989 2.618033988 4.23606798 6.854101955 121393 1.618033989 2.618033989 4.236067976 6.854101971 196418 1.618033989 2.618033989 4.236067978 6.854101965 317811 1.618033989 2.618033989 4.236067977 6.854101967 514229 1.618033989 2.618033989 4.236067978 6.854101966 832040 1.618033989 2.618033989 4.236067977 6.854101966 1346269 1.618033989 2.618033989 4.236067978 6.854101966 2178309 1.618033989 2.618033989 4.236067977 6.854101966 3524578 1.618033989 2.618033989 4.236067978 6.854101966

From the table, we will see that the first column is the value of n, the second column give us the value of the golden ratio (1.61), the third column gives us the square of golden ratio (2.618), and the fourth column gives us  the cube of the golden ratio (4.23) and so on.

Next, we want to see the table when f (0) and f (1) are some arbitrary integers other than 1. If f (0) =1 and f (1) = 3, then our sequence is a Lucas Sequence. All such sequences have the same limit of the ratio of successive terms.

 3 3 4 4 4 7 7 2.333333 7 11 11 2.75 3.666667 11 18 18 2.571429 4.5 6 29 29 2.636364 4.142857 7.25 47 47 2.611111 4.272727 6.714286 76 76 2.62069 4.222222 6.909091 123 123 2.617021 4.241379 6.833333 199 199 2.618421 4.234043 6.862069 322 322 2.617886 4.236842 6.851064 521 521 2.61809 4.235772 6.855263 843 843 2.618012 4.236181 6.853659 1364 1364 2.618042 4.236025 6.854271 2207 2207 2.618031 4.236084 6.854037 3571 3571 2.618035 4.236062 6.854127 5778 5778 2.618034 4.23607 6.854093 9349 9349 2.618034 4.236067 6.854106 15127 15127 2.618034 4.236068 6.854101 24476 24476 2.618034 4.236068 6.854102 39603 39603 2.618034 4.236068 6.854102 64079 64079 2.618034 4.236068 6.854102 103682 103682 2.618034 4.236068 6.854102 167761 167761 2.618034 4.236068 6.854102 271443 271443 2.618034 4.236068 6.854102 439204 439204 2.618034 4.236068 6.854102 710647 710647 2.618034 4.236068 6.854102 1149851 1149851 2.618034 4.236068 6.854102 1860498 1860498 2.618034 4.236068 6.854102

 1 5 5 6 1.2 6 11 1.833333333 2.2 11 17 17 1.545454545 2.833333333 3.4 5.6 28 1.647058824 2.545454545 4.666666667 7.5 45 1.607142857 2.647058824 4.090909091 6.636363636 73 1.622222222 2.607142857 4.294117647 6.941176471 118 1.616438356 2.622222222 4.214285714 6.821428571 191 1.618644068 2.616438356 4.244444444 6.866666667 309 1.617801047 2.618644068 4.232876712 6.849315068 500 1.618122977 2.617801047 4.237288136 6.855932203 809 1.618 2.618122977 4.235602094 6.853403141 1309 1.618046972 2.618 4.236245955 6.854368932 2118 1.61802903 2.618046972 4.236 6.854 3427 1.618035883 2.61802903 4.236093943 6.854140915 5545 1.618033265 2.618035883 4.23605806 6.854087089 8972 1.618034265 2.618033265 4.236071766 6.854107649 14517 1.618033883 2.618034265 4.23606653 6.854099796 23489 1.618034029 2.618033883 4.23606853 6.854102795 38006 1.618033973 2.618034029 4.236067766 6.85410165 61495 1.618033995 2.618033973 4.236068058 6.854102087 99501 1.618033987 2.618033995 4.236067947 6.85410192 160996 1.61803399 2.618033987 4.236067989 6.854101984 260497 1.618033988 2.61803399 4.236067973 6.85410196 421493 1.618033989 2.618033988 4.236067979 6.854101969 681990 1.618033989 2.618033989 4.236067977 0

As n increases,

f (n+1)/f (n) = 1.618, f (n+2) / f (n) = 2.618, f (n +3) / f (n) =4.236, and f (n +4)/ f (n) = 6.854 and so fourth.

From the tables, we can see that the value of n does not determine the outcome of the table/sequences. I find it interesting how the fifth column of the last graph ends up being zero, as n increases the graph will generate a value for this column.