This spreadsheet lists some of the ratios for the first 25 terms of the Fibonacci sequence.

n f(n-1)+f(n-2) Each pair Every other Every third Every fourth Every fifth
0 1 1 2 3 5 8
1 1 2 3 5 8 13
2 2 1.5 2.5 4 6.5 10.5
3 3 1.66666666666667 2.66666666666667 4.33333333333333 7 11.3333333333333
4 5 1.6 2.6 4.2 6.8 11
5 8 1.625 2.625 4.25 6.875 11.125
6 13 1.61538461538462 2.61538461538462 4.23076923076923 6.84615384615385 11.0769230769231
7 21 1.61904761904762 2.61904761904762 4.23809523809524 6.85714285714286 11.0952380952381
8 34 1.61764705882353 2.61764705882353 4.23529411764706 6.85294117647059 11.0882352941176
9 55 1.61818181818182 2.61818181818182 4.23636363636364 6.85454545454545 11.0909090909091
10 89 1.61797752808989 2.61797752808989 4.23595505617978 6.85393258426966 11.0898876404494
11 144 1.61805555555556 2.61805555555556 4.23611111111111 6.85416666666667 11.0902777777778
12 233 1.61802575107296 2.61802575107296 4.23605150214592 6.85407725321888 11.0901287553648
13 377 1.61803713527851 2.61803713527851 4.23607427055703 6.85411140583554 11.0901856763926
14 610 1.61803278688525 2.61803278688525 4.23606557377049 6.85409836065574 11.0901639344262
15 987 1.61803444782168 2.61803444782168 4.23606889564336 6.85410334346505 11.0901722391084
16 1597 1.61803381340013 2.61803381340013 4.23606762680025 6.85410144020038 11.0901690670006
17 2584 1.61803405572755 2.61803405572755 4.23606811145511 6.85410216718266 11.0901702786378
18 4181 1.61803396316671 2.61803396316671 4.23606792633341 6.85410188950012 11.0901698158335
19 6765 1.6180339985218 2.6180339985218 4.23606799704361 6.85410199556541 11.090169992609
20 10946 1.61803398501736 2.61803398501736 4.23606797003472 6.85410195505207 11.0901699250868
21 17711 1.6180339901756 2.6180339901756 4.23606798035119 6.85410197052679
22 28657 1.61803398820532 2.61803398820533 4.23606797641065
23 46368 1.6180339889579 2.6180339889579
24 75025 1.61803398867044
25 121393

It is interesting that in all of these cases, the ratios all approach one common ratio. For instance, for each pair of terms, the common ratio is about 1.681034. This number is called the Golden Ratio and is denoted by the Greek letter:

= 1.681034

The ratio of every other term is given by 1 + this Golden Ratio. Then, an interesting pattern begins to develop. The common ratio of every third term is this number cubed, and for every fourth term, the common ratio is phi raised to the fourth power, etc.

These ratios could be used to predict bigger terms in the Fibonacci sequence. If we wanted to find the 30th term in this sequence, we would multiply 121,393 by 11.09017 because that is the common ratio for when we examine the terms that have a difference of 5. This would show that the 30th term is 1,346,269.

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