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.

Exploring Ratios in the Fibonnaci Sequence

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:

Golden Ratio, GR+1, 2(GR)+1, 3(GR)+2, 5(GR)+3, 8(GR)+5...

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?