Mathematics Problems and explorations with spreadsheets

by

Hyeshin Choi

4. Generate a Fibonnaci sequence in the first column using f(0) = 1, f(1) = 1,

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

a. Construct the ratio of each pair of adjacent terms in the Fibonnaci sequence. What happens as n increases? What about the ratio of every second term? etc.

For the second column, I tried that b2/b1, then as n increases, the even terms are increasing and the odd terms are decreasing. So they have converged to the number 1.61803399 which is gonden ratio.

For the third column, I tried that c3/c11, then as n increases, the even terms are increasing and the odd terms are decreasing. So they are converged to 2.61803399 which is the square of golden ratio.

For the forth column, I tried that d4/d1, then as n increases, the even terms are increasing and the odd terms are decreasing. So they have converged to the number 4.23606798 which is the cube of gonden ratio.

b. Explore sequences where f(0) and f(1) are some arbitrary integers other than 1. If f(0)=1 and f(1) = 4, then your sequence is a Lucas Sequence. All such sequences, however, have the same limit of the ratio of successive terms.

I started with f(1)=1 and f(2)=4, then it is a Lucas sequence. To explore the ratio, I started with differnt number,4 and tried a2/a1 for the second column; however, it is still approaching 1.618033989, which is known as Golden Ratio, just like Fibonnaci Sequence.

Thethird column, I tried a3/a1, and the ratio is still approaching 2.618033989, which is square of Golden Ratio, just like Fibonnaci Sequence.

The forth column, I tried a4/a1, it is still approaching 4.236067977, which is cubic of Golden Ratio, just like Fibonnaci Sequence.