A term in the Fibonacci Sequence is created by adding the previoius two terms in the sequence together. Mathematicians find the sequence interesting because after a while, the ratio of the two last numbers converges to a particular value, called the golden ratio.

Doing all the calculations can be kind of boring, so this web page will let you pick the starting numbers, and it will calculate the sequence for you, so that you can just focus on seering what's the same and what's different when you change things. The equation on the left shows the sum of the previous two terms (can you tell where the numbers come from?) The equation on the right shows the ratio.

The Fibonacci Sequence starts with 1 & 1 as the two starting numbers. What happens if you use the same rules, but make the first number a 0. How about the second number? what happens if you pick two numbers that appear later in the sequence, say 3 & 5? Do parts of the sequence still look familiar? what happens if you use a negative number?

What happens if the two starting numbers don't match any of the numbers in the Fibonacci Sequence? Does that affect what the ratio converges to?

Can you find anything else interesting about this series?

Compare the ratios you found here with the ratios from the Golden Rectangle and the Golden Triangle pages. When you're done with those, you may wonder where that golden ratio comes from, anyway?.

first: second: count:
1 + 1 = 2 2 ÷ 1 = 2
1 + 2 = 3 3 ÷ 2 = 1.5
2 + 3 = 5 5 ÷ 3 = 1.6666666666667
3 + 5 = 8 8 ÷ 5 = 1.6
5 + 8 = 13 13 ÷ 8 = 1.625
8 + 13 = 21 21 ÷ 13 = 1.6153846153846
13 + 21 = 34 34 ÷ 21 = 1.6190476190476
21 + 34 = 55 55 ÷ 34 = 1.6176470588235
34 + 55 = 89 89 ÷ 55 = 1.6181818181818
55 + 89 = 144 144 ÷ 89 = 1.6179775280899
89 + 144 = 233 233 ÷ 144 = 1.6180555555556
144 + 233 = 377 377 ÷ 233 = 1.618025751073
233 + 377 = 610 610 ÷ 377 = 1.6180371352785
377 + 610 = 987 987 ÷ 610 = 1.6180327868852
610 + 987 = 1597 1597 ÷ 987 = 1.6180344478217
987 + 1597 = 2584 2584 ÷ 1597 = 1.6180338134001
1597 + 2584 = 4181 4181 ÷ 2584 = 1.6180340557276
2584 + 4181 = 6765 6765 ÷ 4181 = 1.6180339631667
4181 + 6765 = 10946 10946 ÷ 6765 = 1.6180339985218
6765 + 10946 = 17711 17711 ÷ 10946 = 1.6180339850174
10946 + 17711 = 28657 28657 ÷ 17711 = 1.6180339901756
17711 + 28657 = 46368 46368 ÷ 28657 = 1.6180339882053
28657 + 46368 = 75025 75025 ÷ 46368 = 1.6180339889579
46368 + 75025 = 121393 121393 ÷ 75025 = 1.6180339886704
75025 + 121393 = 196418 196418 ÷ 121393 = 1.6180339887802
121393 + 196418 = 317811 317811 ÷ 196418 = 1.6180339887383
196418 + 317811 = 514229 514229 ÷ 317811 = 1.6180339887543
317811 + 514229 = 832040 832040 ÷ 514229 = 1.6180339887482
514229 + 832040 = 1346269 1346269 ÷ 832040 = 1.6180339887505
832040 + 1346269 = 2178309 2178309 ÷ 1346269 = 1.6180339887496
1346269 + 2178309 = 3524578 3524578 ÷ 2178309 = 1.61803398875
2178309 + 3524578 = 5702887 5702887 ÷ 3524578 = 1.6180339887499
3524578 + 5702887 = 9227465 9227465 ÷ 5702887 = 1.6180339887499
5702887 + 9227465 = 14930352 14930352 ÷ 9227465 = 1.6180339887499
9227465 + 14930352 = 24157817 24157817 ÷ 14930352 = 1.6180339887499
14930352 + 24157817 = 39088169 39088169 ÷ 24157817 = 1.6180339887499
24157817 + 39088169 = 63245986 63245986 ÷ 39088169 = 1.6180339887499
39088169 + 63245986 = 102334155 102334155 ÷ 63245986 = 1.6180339887499
63245986 + 102334155 = 165580141 165580141 ÷ 102334155 = 1.6180339887499
102334155 + 165580141 = 267914296 267914296 ÷ 165580141 = 1.6180339887499