University of Georgia

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Fibonacci and the Golden Ratio

The relationship between the Fibonacci Sequence and the Golden Ratio is a surprising one. We have two seemingly unrelated topics producing the same exact number. Considering that this number (or Golden Ratio) is non-rational, the occurance is beyond coincidence. It calls for futher examination...

The Golden Ratio = (sqrt(5) + 1)/2 or about 1.618

The Golden Ratio is, perhaps, best visually displayed in the Golden Rectangle. This rectangle has the property that its length is in Golen Ratio with its width. As a consequence, we can divide this rectangle into a square and a smaller rectangle that is similar to the first. Let the following GSP sketch illustrate: The Fibonacci Sequence is one where each term is defined as the sum of the two previous terms: We can create this sequence easily in a spreadsheet, using the formula above. This has been done in the center column of the spreadsheet below:

 1 1 2 1 1 3 2 2 4 3 1.5 5 5 1.66666666666667 6 8 1.6 7 13 1.625 8 21 1.61538461538462 9 34 1.61904761904762 10 55 1.61764705882353 11 89 1.61818181818182 12 144 1.61797752808989 13 233 1.61805555555556 14 377 1.61802575107296 15 610 1.61803713527851 16 987 1.61803278688525 17 1597 1.61803444782168 18 2584 1.61803381340013 19 4181 1.61803405572755 20 6765 1.61803396316671 21 10946 1.6180339985218 22 17711 1.61803398501736 23 28657 1.6180339901756 24 46368 1.61803398820533 25 75025 1.6180339889579 26 121393 1.61803398867044 27 196418 1.61803398878024 28 317811 1.6180339887383 29 514229 1.61803398875432 30 832040 1.6180339887482 31 1346269 1.61803398875054 32 2178309 1.61803398874965 33 3524578 1.61803398874999 34 5702887 1.61803398874986 35 9227465 1.61803398874991 36 14930352 1.61803398874989 37 24157817 1.6180339887499 38 39088169 1.61803398874989 39 63245986 1.6180339887499 40 102334155 1.61803398874989 41 165580141 1.6180339887499 42 267914296 1.6180339887499 43 433494437 1.6180339887499 44 701408733 1.6180339887499 45 1134903170 1.6180339887499 46 1836311903 1.6180339887499 47 2971215073 1.6180339887499 48 4807526976 1.6180339887499 49 7778742049 1.6180339887499 50 12586269025 1.6180339887499

We have also taken the ratio of every two consecutive terms, in the right column. If we take the limit of this ratio as the terms get larger... the Golden Ratio!

Understanding the Relationship

Remember, now, that the Golden Rectangle can be divided into a square and another Golden Rectangle. In fact, we can repeat this process time and again. Suppose we start with a rectangle with length Y and width X. When we divide this rectangle into a square and a new Golden Rectangle, we get length X and width Y-X for the new rectangle. Since the sides of the rectangles are in the same proportion (namely 1.618...), we get: Now, let's return to the Fibonacci Sequence. The ratio of consecutive terms forms a sequence itself. Suppose we know that this sequence converges to some real number (we can prove this in a lesson on geometric sequences). Then, for large values of n: But , so And, if we replace with Y and with X, we have The Golen Ratio!