# By Donna Greenwood

The spreadsheet is a utility tool that can be adapted to many different explorations, presentations and simulations in mathematics. This write up explores the Fibonacci sequence using Excel.

 Fibonacci Numbers Ratio of terms Ratio of 2nd terms Ratio of 3rd terms Ratio of 4th terms Ratio of 5th terms 1 1 1 2 2 2 3 1.5 3 3 5 1.666666667 2.5 5 5 8 1.6 2.66666667 4 8 8 13 1.625 2.6 4.33333333 6.5 13 21 1.615384615 2.625 4.2 7 10.5 34 1.619047619 2.61538462 4.25 6.8 11.3333333 55 1.617647059 2.61904762 4.23076923 6.875 11 89 1.618181818 2.61764706 4.23809524 6.84615385 11.125 144 1.617977528 2.61818182 4.23529412 6.85714286 11.0769231 233 1.618055556 2.61797753 4.23636364 6.85294118 11.0952381 377 1.618025751 2.61805556 4.23595506 6.85454545 11.0882353 610 1.618037135 2.61802575 4.23611111 6.85393258 11.0909091 987 1.618032787 2.61803714 4.2360515 6.85416667 11.0898876 1597 1.618034448 2.61803279 4.23607427 6.85407725 11.0902778 2584 1.618033813 2.61803445 4.23606557 6.85411141 11.0901288 4181 1.618034056 2.61803381 4.2360689 6.85409836 11.0901857 6765 1.618033963 2.61803406 4.23606763 6.85410334 11.0901639 10946 1.618033999 2.61803396 4.23606811 6.85410144 11.0901722 17711 1.618033985 2.618034 4.23606793 6.85410217 11.0901691 28657 1.61803399 2.61803399 4.236068 6.85410189 11.0901703 46368 1.618033988 2.61803399 4.23606797 6.854102 11.0901698 75025 1.618033989 2.61803399 4.23606798 6.85410196 11.09017 121393 1.618033989 2.61803399 4.23606798 6.85410197 11.0901699 196418 1.618033989 2.61803399 4.23606798 6.85410196 11.09017 317811 1.618033989 2.61803399 4.23606798 6.85410197 11.0901699 514229 1.618033989 2.61803399 4.23606798 6.85410197 11.0901699

## ; the ratios of the second terms (i.e. every other term) converge to 1G + 1; the ratios of the third terms converge to 2G+1; the ratios of the fourth terms converge to 3G+1 and the ratios of the fifth to 4G+1.

Here’s a graph of all the ratios:

By graphing these together, you can see the similar shape of the graphs and how quickly each converges to its particular value.

Next, look at the Lucas sequence. This sequence starts with (2), 1 and 3 where the Fibonacci sequence starts with (0), 1 and 1.

 Lucas Sequence Ratio of terms Ratio of 2nd terms Ratio of 3rd terms Ratio of 4th terms Ratio of 5th terms 1 3 3 4 1.333333333 4 7 1.75 2.333333 7 11 1.571428571 2.75 3.666667 11 18 1.636363636 2.571429 4.5 6 18 29 1.611111111 2.636364 4.142857 7.25 9.666666667 47 1.620689655 2.611111 4.272727 6.714285714 11.75 76 1.617021277 2.62069 4.222222 6.909090909 10.85714286 123 1.618421053 2.617021 4.241379 6.833333333 11.18181818 199 1.617886179 2.618421 4.234043 6.862068966 11.05555556 322 1.618090452 2.617886 4.236842 6.85106383 11.10344828 521 1.618012422 2.61809 4.235772 6.855263158 11.08510638 843 1.618042226 2.618012 4.236181 6.853658537 11.09210526 1364 1.618030842 2.618042 4.236025 6.854271357 11.08943089 2207 1.618035191 2.618031 4.236084 6.854037267 11.09045226 3571 1.61803353 2.618035 4.236062 6.854126679 11.09006211 5778 1.618034164 2.618034 4.23607 6.854092527 11.09021113 9349 1.618033922 2.618034 4.236067 6.854105572 11.09015421 15127 1.618034014 2.618034 4.236068 6.854100589 11.09017595 24476 1.618033979 2.618034 4.236068 6.854102492 11.09016765 39603 1.618033992 2.618034 4.236068 6.854101765 11.09017082 64079 1.618033987 2.618034 4.236068 6.854102043 11.09016961 103682 1.618033989 2.618034 4.236068 6.854101937 11.09017007 167761 1.618033989 2.618034 4.236068 6.854101977 11.09016989 271443 1.618033989 2.618034 4.236068 6.854101962 11.09016996 439204 1.618033989 2.618034 4.236068 6.854101968 11.09016994 710647 1.618033989 2.618034 4.236068 6.854101966 11.09016995 1149851 1.618033989 2.618034 4.236068 6.854101966 11.09016994

This convergence of the ratios is the same as the Fibonacci sequence.

Finally, here’s a random sequence beginning with -1 and -7:…

 Random Sequence Ratio of terms Ratio of 2nd terms Ratio of 3rd terms Ratio of 4th terms Ratio of 5th terms -1 -7 7 -8 1.142857143 8 -15 1.875 2.142857 15 -23 1.533333333 2.875 3.285714 23 -38 1.652173913 2.533333 4.75 5.428571 38 -61 1.605263158 2.652174 4.066667 7.625 8.714286 -99 1.62295082 2.605263 4.304348 6.6 12.375 -160 1.616161616 2.622951 4.210526 6.956522 10.66667 -259 1.61875 2.616162 4.245902 6.815789 11.26087 -419 1.617760618 2.61875 4.232323 6.868852 11.02632 -678 1.618138425 2.617761 4.2375 6.848485 11.11475 -1097 1.6179941 2.618138 4.235521 6.85625 11.08081 -1775 1.618049225 2.617994 4.236277 6.853282 11.09375 -2872 1.618028169 2.618049 4.235988 6.854415 11.0888 -4647 1.618036212 2.618028 4.236098 6.853982 11.09069 -7519 1.61803314 2.618036 4.236056 6.854148 11.08997 -12166 1.618034313 2.618033 4.236072 6.854085 11.09025 -19685 1.618033865 2.618034 4.236066 6.854109 11.09014 -31851 1.618034036 2.618034 4.236069 6.854099 11.09018 -51536 1.618033971 2.618034 4.236068 6.854103 11.09017 -83387 1.618033996 2.618034 4.236068 6.854102 11.09017 -134923 1.618033986 2.618034 4.236068 6.854102 11.09017 -218310 1.61803399 2.618034 4.236068 6.854102 11.09017 -353233 1.618033988 2.618034 4.236068 6.854102 11.09017 -571543 1.618033989 2.618034 4.236068 6.854102 11.09017 -924776 1.618033989 2.618034 4.236068 6.854102 11.09017 -1496319 1.618033989 2.618034 4.236068 6.854102 11.09017 -2421095 1.618033989 2.618034 4.236068 6.854102 11.09017

Here’s a comparison of the convergence of the first terms of the three sequences listed here:

After the initial jumping around to account for the differences in the first terms, the terms converge rapidly – by about the 6th number in the sequence there’s no measurable difference in the ratios, at least as far as the graph is concerned.

Last but not least, I wanted to compare the recursive formula – an = an-1 + an – to the closed form of the equation, which is

an = 1/sqrt(5) * (((1 + sqrt(5))/2)n – ((1 - sqrt(5))/2)n). I expected to see some round off error based on the way the spreadsheet would handle the square roots raised to a power. Here are the results:

 Fibonacci Numbers Closed Formula 1 1 1 1 2 2 3 3 5 5 8 8 13 13 21 21 34 34 55 55 89 89 144 144 233 233 377 377 610 610 987 987 1597 1597 2584 2584 4181 4181 6765 6765 10946 10946 17711 17711 28657 28657 46368 46368 75025 75025 121393 121393 196418 196418 317811 317811 514229 514229

As you can see, there’s not a bit of error! That was a nice surprise, especially since this was the formula I had to key into the spreadsheet:

=1/SQRT(5)*(POWER((1+SQRT(5))/2,B2) - POWER((1-SQRT(5))/ 2, B2))

Note: B2 is a hidden value that represents n in the prior equation.