Becky Dragan

Assignment 12

The Spreadsheet in Mathematics Explorations

The spreadsheet can be used for many different things. In this assignment, I will explore how EXCEL can be used to graph any function y = f(x).

Let's look at a graph of y = x^2 + x + 1 for x=-15 to 15. In EXCEL the x-values can be entered into the first column of cells and the y-values in the second column. The function can then be graphed and would look as follows.

Other functions such as cosine and sine can be graphed as well. A graph of y = cos(0.5x) would look as follows.

The spreadsheet can also be used to examine Fibonnaci sequence. The first column contains the Fibonnaci numbers using f(0)=1, f(1)=1, f(n)=f(n-1)+f(n-2). The second column is the ratio of each adjacent pair of terms in the Fibonnaci sequence. The third column is the ratio of every second term, the fourth column is the ratio of every third term, and the fifth column is the ratio of every fourth term. What do you notice about these ratios? Can you predict what would happen if you took the ratio of every fifth term?

 1 1 1 2 0.5 0.5 3 0.666666666666667 0.333333333333333 0.333333333333333 5 0.6 0.4 0.2 0.2 8 0.625 0.375 0.25 0.125 13 0.615384615384615 0.384615384615385 0.230769230769231 0.153846153846154 21 0.619047619047619 0.380952380952381 0.238095238095238 0.142857142857143 34 0.617647058823529 0.382352941176471 0.235294117647059 0.147058823529412 55 0.618181818181818 0.381818181818182 0.236363636363636 0.145454545454545 89 0.617977528089888 0.382022471910112 0.235955056179775 0.146067415730337 144 0.618055555555556 0.381944444444444 0.236111111111111 0.145833333333333 233 0.618025751072961 0.381974248927039 0.236051502145923 0.145922746781116 377 0.618037135278515 0.381962864721485 0.236074270557029 0.145888594164456 610 0.618032786885246 0.381967213114754 0.236065573770492 0.145901639344262 987 0.618034447821682 0.381965552178318 0.236068895643364 0.145896656534954 1597 0.618033813400125 0.381966186599875 0.23606762680025 0.145898559799624 2584 0.618034055727554 0.381965944272446 0.236068111455108 0.145897832817337 4181 0.618033963166707 0.381966036833293 0.236067926333413 0.14589811049988 6765 0.618033998521803 0.381966001478197 0.236067997043607 0.14589800443459 10946 0.618033985017358 0.381966014982642 0.236067970034716 0.145898044947926 17711 0.618033990175597 0.381966009824403 0.236067980351194 0.145898029473209 28657 0.618033988205325 0.381966011794675 0.23606797641065 0.145898035384025 46368 0.618033988957902 0.381966011042098 0.236067977915804 0.145898033126294 75025 0.618033988670443 0.381966011329557 0.236067977340886 0.14589803398867 121393 0.618033988780243 0.381966011219757 0.236067977560485 0.145898033659272 196418 0.618033988738303 0.381966011261697 0.236067977476606 0.145898033785091 317811 0.618033988754323 0.381966011245677 0.236067977508645 0.145898033737032 514229 0.618033988748204 0.381966011251796 0.236067977496407 0.145898033755389 832040 0.618033988750541 0.381966011249459 0.236067977501082 0.145898033748377

You can do other sequences where f(0) and f(1) are arbitrary integers. Let's look at the sequence where f(0) = 5 and f(1) = 13. Each successsive term is found in the same manner as above f(n) = f(n-1) + f(n-2). In the chart below, the first column is the sequence, the second column is the ratio of to adjacent terms, and the third column is the ratio of every second term. What do you notice about this chart compared to the Fibonnaci sequence? Each ratio has the same limit. Do you think this is true for all such sequences?

 5 13 0.384615384615385 18 0.722222222222222 0.277777777777778 31 0.580645161290323 0.419354838709677 49 0.63265306122449 0.36734693877551 80 0.6125 0.3875 129 0.62015503875969 0.37984496124031 209 0.617224880382775 0.382775119617225 338 0.618343195266272 0.381656804733728 547 0.617915904936015 0.382084095063985 885 0.618079096045198 0.381920903954802 1432 0.618016759776536 0.381983240223464 2317 0.618040569702201 0.381959430297799 3749 0.618031475060016 0.381968524939984 6066 0.618034948895483 0.381965051104517 9815 0.618033622007132 0.381966377992868 15881 0.618034128833197 0.381965871166803 25696 0.618033935242839 0.381966064757161 41577 0.618034009187772 0.381965990812228 67273 0.618033980943321 0.381966019056679