If we look at the ratio for every second term of the Fibonnaci Sequence, we get the following data for the first 41 terms.
| 1 | |
| 1 | 1 |
| 2 | 2 |
| 3 | 1.5 |
| 5 | 1.66666666666667 |
| 8 | 1.6 |
| 13 | 1.625 |
| 21 | 1.61538461538462 |
| 34 | 1.61904761904762 |
| 55 | 1.61764705882353 |
| 89 | 1.61818181818182 |
| 144 | 1.61797752808989 |
| 233 | 1.61805555555556 |
| 377 | 1.61802575107296 |
| 610 | 1.61803713527851 |
| 987 | 1.61803278688525 |
| 1597 | 1.61803444782168 |
| 2584 | 1.61803381340013 |
| 4181 | 1.61803405572755 |
| 6765 | 1.61803396316671 |
| 10946 | 1.6180339985218 |
| 17711 | 1.61803398501736 |
| 28657 | 1.6180339901756 |
| 46368 | 1.61803398820532 |
| 75025 | 1.6180339889579 |
| 121393 | 1.61803398867044 |
| 196418 | 1.61803398878024 |
| 317811 | 1.6180339887383 |
| 514229 | 1.61803398875432 |
| 832040 | 1.6180339887482 |
| 1346269 | 1.61803398875054 |
| 2178309 | 1.61803398874965 |
| 3524578 | 1.61803398874999 |
| 5702887 | 1.61803398874986 |
| 9227465 | 1.61803398874991 |
| 14930352 | 1.61803398874989 |
| 24157817 | 1.6180339887499 |
| 39088169 | 1.61803398874989 |
| 63245986 | 1.6180339887499 |
| 102334155 | 1.61803398874989 |
| 165580141 | 1.61803398874989 |
The ratio seems to be reaching a limit that is the golden ratio. If we compare this to the data for the ratios for every third, fourth and fifth term, we get the following spreadsheet.
Were you able to determine the pattern that develops with respect to the limits of each of these ratios? We can determine subsequent limits of ratios of terms in two ways. If you notice at the bottom of the spreadsheet above, there is the following notation:
Notice that the "coefficient" of the golden ratio and the "constant" term in these expressions are each Fibonnaci Sequences. We can easily use this pattern to find subsequent terms and thus find the ratios of every eighth, nineth... term.
Notice also that we can find subsequent terms (limits of subsequent ratios) simply by summing the two previous terms. For example, the limit of the ratio of every eighth term will be (8(GR)+5)+(13(GR)+8)=21(GR)+13 which is approximately 46.97871376 (for GR estimated to be 1.618033989).
Build your own EXCEL file to verify this. Can you find the limit for the ratio of every 13th term without finding each of the previous terms?