Fall 2000

Assignment #12

Problem # 4

This write-up investigates the Fibonnaci Sequence where f(0) = 0, f(1) = 1, and f(n) = f(n-1) + F(n-2). There is also the Lucas Sequence where f(0) = 1 and f(1) = 3. You can also use Microsoft Excel to exlore other sequences that begin with any two integers you choose. Observe the charts of the sequences below and return to the disscussion.

The ratio of each pair of adjacent terms is given in the column labeled ratio first. Notice that the ratios converge to the Golden Ratio = 1.61803398874965.

The golden ratio is denoted as f. The exact value of the golden ratio is given below:

.

Here is a small proof that Dr. Wilson gave me to prove this:

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Look at the limits of the ratios of every second, third, fourth, fifth, and sixth terms in the chart.

Here is a summary of the limits of these ratios:

Second

Third

Fourth

Fifth

Sixth

The proofs of the limits can be done using the proof above:

The proofs of the eqivalences of the powers of F begin a little like this:

The other equivalences can be proven in a similar fashion.

OTHER WRITE-UPS

Fibonnaci Sequence

 sequence ratio ratio ratio ratio ratio ratio first second third fourth fifth sixth 1 1 1 2 2 2 3 1.5 3 3 5 1.66666666666667 2.5 5 5 8 1.6 2.66666666666667 4 8 8 13 1.625 2.6 4.33333333333333 6.5 13 13 21 1.61538461538462 2.625 4.2 7 10.5 21 34 1.61904761904762 2.61538461538462 4.25 6.8 11.3333333333333 17 55 1.61764705882353 2.61904761904762 4.23076923076923 6.875 11 18.3333333333333 89 1.61818181818182 2.61764705882353 4.23809523809524 6.84615384615385 11.125 17.8 144 1.61797752808989 2.61818181818182 4.23529411764706 6.85714285714286 11.0769230769231 18 233 1.61805555555556 2.61797752808989 4.23636363636364 6.85294117647059 11.0952380952381 17.9230769230769 377 1.61802575107296 2.61805555555556 4.23595505617978 6.85454545454545 11.0882352941176 17.952380952381 610 1.61803713527851 2.61802575107296 4.23611111111111 6.85393258426966 11.0909090909091 17.9411764705882 987 1.61803278688525 2.61803713527851 4.23605150214592 6.85416666666667 11.0898876404494 17.9454545454545 1597 1.61803444782168 2.61803278688525 4.23607427055703 6.85407725321888 11.0902777777778 17.9438202247191 2584 1.61803381340013 2.61803444782168 4.23606557377049 6.85411140583554 11.0901287553648 17.9444444444444 4181 1.61803405572755 2.61803381340013 4.23606889564336 6.85409836065574 11.0901856763926 17.9442060085837 6765 1.61803396316671 2.61803405572755 4.23606762680025 6.85410334346505 11.0901639344262 17.9442970822281 10946 1.6180339985218 2.61803396316671 4.23606811145511 6.85410144020038 11.0901722391084 17.944262295082 17711 1.61803398501736 2.6180339985218 4.23606792633341 6.85410216718266 11.0901690670006 17.9442755825735 28657 1.6180339901756 2.61803398501736 4.23606799704361 6.85410188950012 11.0901702786378 17.944270507201 46368 1.61803398820532 2.6180339901756 4.23606797003472 6.85410199556541 11.0901698158335 17.9442724458204 75025 1.6180339889579 2.61803398820533 4.23606798035119 6.85410195505207 11.090169992609 17.9442717053337 121393 1.61803398867044 2.6180339889579 4.23606797641065 6.85410197052679 11.0901699250868 17.9442719881744 196418 1.61803398878024 2.61803398867044 4.2360679779158 6.85410196461598 11.090169950878 17.9442718801389 317811 1.6180339887383 2.61803398878024 4.23606797734089 6.85410196687371 11.0901699410266 17.9442719214048 514229 1.61803398875432 2.6180339887383 4.23606797756049 6.85410196601133 11.0901699447895 17.9442719056426 832040 1.6180339887482 2.61803398875432 4.23606797747661 6.85410196634073 11.0901699433522 17.9442719116632 1346269 1.61803398875054 2.6180339887482 4.23606797750864 6.85410196621491 11.0901699439012 17.9442719093635 2178309 1.61803398874965 2.61803398875054 4.23606797749641 6.85410196626297 11.0901699436915 17.9442719102419