## EMAT 6680 Assignment #12

#### By Kent Wiginton

We will be working with Microsoft Excel in this assignment. Specifically we are going to explore the Fibonnaci seqence and see what we can discover.

The Fibonnaci sequence is a sequence that starts with f(0) = 1 and f(1) = 1. The formula for constucting the rest of the sequence is

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

We will construct the sequence and then we will look at the differences between the adjacent and every second term. Also we will look at other sequences of this type by changing f(0) and f(1).

We can put in the inital values into Excel and then put our formula into the next cell and drag down to get Excel to figure out the rest. Next I compared the ratios of adjacent terms. We can do this just as before. This time in the first cell in the column B put the fomula for the ratio of the two previous terms. And then in the next column I compared the ratio of every second pair.

 Sequence Ratio of adjacent pair Ratio of every second pair 1 1 1 2 2 2 3 1.5 3 5 1.66666666666667 2.5 8 1.6 2.66666666666667 13 1.625 2.6 21 1.61538461538462 2.625 34 1.61904761904762 2.61538461538462 55 1.61764705882353 2.61904761904762 89 1.61818181818182 2.61764705882353 144 1.61797752808989 2.61818181818182 233 1.61805555555556 2.61797752808989 377 1.61802575107296 2.61805555555556 610 1.61803713527851 2.61802575107296 987 1.61803278688525 2.61803713527851 1597 1.61803444782168 2.61803278688525 2584 1.61803381340013 2.61803444782168 4181 1.61803405572755 2.61803381340013 6765 1.61803396316671 2.61803405572755

Here we looked at different sequences by changing the inital value of f(0) and f(1).

 f(0)=1,f(1)=3 Ratio of adjacent pair f(0)=4, f(1)=8 Ratio of adjacent pair f(0)=5, f(1)=3 Ratio of adjacent pairs 1 4 5 3 3 8 2 3 0.6 4 1.33333333333333 12 1.5 8 2.66666666666667 7 1.75 20 1.66666666666667 11 1.375 11 1.57142857142857 32 1.6 19 1.72727272727273 18 1.63636363636364 52 1.625 30 1.57894736842105 29 1.61111111111111 84 1.61538461538462 49 1.63333333333333 47 1.62068965517241 136 1.61904761904762 79 1.61224489795918 76 1.61702127659574 220 1.61764705882353 128 1.62025316455696 123 1.61842105263158 356 1.61818181818182 207 1.6171875 199 1.61788617886179 576 1.61797752808989 335 1.61835748792271 322 1.61809045226131 932 1.61805555555556 542 1.61791044776119 521 1.61801242236025 1508 1.61802575107296 877 1.61808118081181 843 1.61804222648752 2440 1.61803713527851 1419 1.61801596351197 1364 1.61803084223013 3948 1.61803278688525 2296 1.61804087385483 2207 1.61803519061584 6388 1.61803444782168 3715 1.61803135888502 3571 1.6180335296783 10336 1.61803381340013 6011 1.61803499327052 5778 1.61803416409969 16724 1.61803405572755 9726 1.61803360505739 9349 1.61803392177224 27060 1.61803396316671 15737 1.61803413530742 15127 1.61803401433308 43784 1.6180339985218 25463 1.61803393276991

Click here to see an Excel worksheet that give the desired data.

Notice that the ratio of each adjacent pair is converging to the golden ratio. And when we take every second ratio that number is converging to 1 + the golden ratio.

Try it for yourself. Click here to try to constuct the Fibonnaci sequence and the golden ration with Excel.