# The Fibonacci Sequence

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The Fibonacci Sequence

In Liber Abaci, a problem is posed that gives rise to the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on to infinity, known today as the Fibonacci sequence. The problem is this:

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How many pairs of rabbits placed in an enclosed area can be produced in a single year from one pair of rabbits if each pair gives birth to a new pair each month starting with the second month?

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In arriving at the solution, we find that each pair, including the first pair, needs a month’s time to mature, but once in production, begets a new pair each month. The number of pairs is the same at the beginning of each of the first two months, so the sequence is 1, 1. This first pair finally doubles its number during the second month, so that there are two pairs at the beginning of the third month. Of these, the older pair begets a third pair the following month so that at the beginning of the fourth month, the sequence expands 1, 1, 2, 3. Of these three, the two older pairs reproduce, but not the youngest pair, so the number of rabbit pairs expands to five. The next month, three pairs reproduce so the sequence expands to 1, 1, 2, 3, 5, 8 and so forth. Figure 3-1 shows the Rabbit Family Tree with the family growing with exponential acceleration. Continue the sequence for a few years and the numbers become astronomical. In 100 months, for instance, we would have to contend with 354,224,848,179,261,915,075 pairs of rabbits. The Fibonacci sequence resulting from the rabbit problem has many interesting properties and reflects an almost constant relationship among its components.

Figure 3-1

The sum of any two adjacent numbers in the sequence forms the next higher number in the sequence, viz., 1 plus 1 equals 2, 1 plus 2 equals 3, 2 plus 3 equals 5, 3 plus 5 equals 8, and so on to infinity.

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The Golden Ratio

After the first several numbers in the sequence, the ratio of any number to the next higher is approximately .618 to 1 and to the next lower number approximately 1.618 to 1. The further along the sequence, the closer the ratio approaches phi (denoted ϕ) which is an irrational number, .618034…. Between alternate numbers in the sequence, the ratio is approximately .382, whose inverse is 2.618. Refer to Figure 3-2 for a ratio table interlocking all Fibonacci numbers from 1 to 144.

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Phi is the only number that when added to 1 yields its inverse: 1 + .618 = 1 ÷ .618. This alliance of the additive and the multiplicative produces the following sequence of equations:

.6182 = 1 – .618,

.6183 = .618 – .6182,

.6184 = .6182 – .6183,

.6185 = .6183 – .6184, etc.

or alternatively,

1.6182 = 1 + 1.618,

1.6183 = 1.618 + 1.6182,

1.6184 = 1.6182 + 1.6183,

1.61855 = 1.6183 + 1.6184, etc.

Some statements of the interrelated properties of these four main ratios can be listed as follows:

1.618 – .618 = 1,

1.618 x .618 = 1,

1 – .618 = .382,

.618 x .618 = .382,

2.618 – 1.618 = 1,

2.618 x .382 = 1,

2.618 x .618 = 1.618,

1.618 x 1.618 = 2.618.

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Besides 1 and 2, any Fibonacci number multiplied by four, when added to a selected Fibonacci number, gives another Fibonacci number, so that:

Figure 3-2

3 x 4 = 12; + 1 = 13,

5 x 4 = 20; + 1 = 21,

8 x 4 = 32; + 2 = 34,

13 x 4 = 52; + 3 = 55,

21 x 4 = 84; + 5 = 89, and so on.

As the new sequence progresses, a third sequence begins in those numbers that are added to the 4x multiple. This relationship is possible because the ratio between second alternate Fibonacci numbers is 4.236, where .236 is both its inverse and its difference from the number 4. Other multiples produce different sequences, all based on Fibonacci multiples.

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We offer a partial list of additional phenomena relating to the Fibonacci sequence as follows:

1) No two consecutive Fibonacci numbers have any common factors.

2) If the terms of the Fibonacci sequence are numbered 1, 2, 3, 4, 5, 6, 7, etc., we find that, except for the fourth Fibonacci number (3), each time a prime Fibonacci number (one divisible only by itself and 1) is reached, the sequence number is prime as well. Similarly, except for the fourth Fibonacci number (3), all composite sequence numbers (those divisible by at least two numbers besides themselves and 1) denote composite Fibonacci numbers, as in the table below. The converses of these phenomena are not always true.

Fibonacci: Prime vs. Composite

P P P X P P P P

1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

X C C C C C C C C

3) The sum of any ten numbers in the sequence is divisible by 11.

4) The sum of all Fibonacci numbers in the sequence up to any point, plus 1, equals the Fibonacci number two steps ahead of the last one

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