EMAT 6680 Assignment 12

Mathematic problems and explorations with spreadsheets.

Melissa Bauers Summer 2002

Generate a Fibonnaci Sequence

The Fibonnaci Sequence is the sequence of 1, 1, 2, 3, 5, 8, 13, … It is generated by taking the sum of the two previous terms. The equation is f(n) = f(n-1) + f(n-2).

This is a example of a Fibonnaci Square. Each color represents a number in the sequence. In this assignment, we will look at the Fibonnaci Sequence using a spreadsheet.

 n f(n)=f(n-1) + f(n-2) 0 1 1 1 2 2 3 3 4 5 5 8 6 13 7 21 8 34 9 55 10 89 11 144 12 233 13 377 14 610 15 987 16 1597 17 2584 18 4181 19 6765 20 10946 21 17711 22 28657 23 46368 24 75025 25 121393 26 196418 27 317811 28 514229 29 832040 30 1346269 31 2178309 32 3524578 33 5702887 34 9227465 35 14930352 36 24157817 37 39088169 38 63245986

As n increases the f(n) becomes increasingly bigger. Notice the sharp curve of the graph. Now let's look at the ratio of the second term compared to the first.

Now let us examine the ratio of terms.

 n f(n)=f(n-1) + f(n-2) Ratio of f(n)/f(n-1) Ratio of f(n)/f(n-2) Ratio of f(n-1)/f(n-2) 0 1 1 1 1 2 2 2 2 1 3 3 1.5 3 2 4 5 1.666666667 2.5 1.5 5 8 1.6 2.66666667 1.666666669 6 13 1.625 2.6 1.6 7 21 1.615384615 2.625 1.625 8 34 1.619047619 2.615384615 1.615384615 9 55 1.617647059 2.619047619 1.619047619 10 89 1.618181818 2.617647059 1.617647059 11 144 1.617977528 2.618181818 1.618181818 12 233 1.618055556 2.617977528 1.617977528 13 377 1.618025751 2.618055556 1.618055556 14 610 1.618037135 2.618025751 1.618025751 15 987 1.618032787 2.618037135 1.618037135 16 1597 1.618034448 2.618032787 1.618032787 17 2584 1.618033813 2.618034448 1.618034448 18 4181 1.618034056 2.618033813 1.618033813 19 6765 1.618033963 2.618034056 1.618034056 20 10946 1.618033999 2.618033963 1.618033963 21 17711 1.618033985 2.618033999 1.618033999 22 28657 1.61803399 2.618033985 1.618033985 23 46368 1.618033988 2.61803399 1.61803399 24 75025 1.618033989 2.618033988 1.618033988 25 121393 1.618033989 2.618033989 1.618033989

Notice the ratio of terms in column 3. They come out to be the golden ration. The ratios in columns 4 and 5 follow suit.

The next series will start out with different values for f(0) and f(1). The Lucas Sequence starts as f(0) = 1 and f(1) =3. Now look at the ratios.

 n f(n)=f(n-1) + f(n-2) Ratio of f(n)/f(n-1) Ratio of f(n)/f(n-2) Ratio of f(n-1)/f(n-2) 0 1 1 3 3 2 4 1.333333333 4 3.000000001 3 7 1.75 2.33333333 1.333333331 4 11 1.571428571 2.75 1.75 5 18 1.636363636 2.571428571 1.571428571 6 29 1.611111111 2.636363636 1.636363636 7 47 1.620689655 2.611111111 1.611111111 8 76 1.617021277 2.620689655 1.620689655 9 123 1.618421053 2.617021277 1.617021277 10 199 1.617886179 2.618421053 1.618421053 11 322 1.618090452 2.617886179 1.617886179 12 521 1.618012422 2.618090452 1.618090452 13 843 1.618042226 2.618012422 1.618012422 14 1364 1.618030842 2.618042226 1.618042226 15 2207 1.618035191 2.618030842 1.618030842 16 3571 1.61803353 2.618035191 1.618035191 17 5778 1.618034164 2.61803353 1.61803353 18 9349 1.618033922 2.618034164 1.618034164 19 15127 1.618034014 2.618034014 1.618033979 20 24476 1.618033979 2.618033979 1.618033992 21 39603 1.618033992 2.618033992 1.618033987 22 64079 1.618033987 2.618033987 1.618033989 23 103682 1.618033989 2.618033989 1.618033988 24 167761 1.618033989 2.618033989 1.618033989 25 271443 1.618033989 2.618033989 1.618033989

The ratios converge to the golden ratio. This convergence does not require f(n) to start at a certain value.