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
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.