Fibonacci sequence

The Fibonacci sequence is a naturally occurring phenomena in nature. It was discovered by Leonardo Fibonacci. Leonardo was an Italian mathematician who lived from about 1180 to about 1250 CE.

Mathematicians today are still finding interesting way this series of numbers describes nature. To see how this sequence describes nature, take a close look at the figure below:
Equiangular spiral
This spiral shape is found in many flowers, pine cones, and snails' shell to mention just a few

What exactly is happening here as far as math is concerned?


You can see that we begin with two squares with a side length that is equal to 1

Then, to get the side length of the third square, we add the side lengths of the two previous squares that is 1 and 1 ( 1 + 1 = 2)

To get the side length of a fourth square, we add 1 and 2 ( 1 + 2 = 3)

To get the side length of a fifth square, we add 2 and 3 ( 2 + 3 = 5)

If we continue this pattern we get:

3 + 5 = 8

5 + 8 = 13

8 + 13 = 21

13 + 21 = 34

21 + 34 = 55

34 + 55 = 89

55 + 89 = 144

89 + 144 = 233


Here is a short list of the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233

Each number in the sequence is the sum of the two numbers before it

We can try to derive a Fibonacci sequence formula by making some observations

F1 = 1

F2 = 1

F3 = F2 + F1 = 1 + 1 = 2

F4 = F3 + F2 = 2 + 1 = 3

F5 = F4 + F3 = 3 + 2 = 5

F6 = F6-1 + F6-2 = F5 + F4 = 5 + 3 = 8

F7 = F7-1 + F7-2 = 8 + 5 = 13

......

......

......

Fn = Fn-1 + Fn-2

Try this

Find the sum of the first ten terms of the Fibonacci sequence

1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143

Now, we will choose numbers other than 1 and 1 to create other Fibonacci-like sequences

2, 2 , 4, 6, 10, 16, 26, 42, 68, 110

The sum is 2 + 2 + 4 + 6 + 10 + 16 + 26 + 42 + 68 + 110 = 286

what if we start with 3 and 3?

3, 3, 6, 9, 15, 24, 39, 63, 102, 165

3 + 3 + 6 + 9 + 15 + 24 + 39 + 63 + 102 + 165 = 429

Now, we shall make a nice observation?

143/11 = 13

286/11 = 26

429/11 = 39

143 = 11 × 13 = 11 × 13 × 1

286 = 11 × 26 = 11 × 13 × 2

429 = 11 × 39 = 11 × 13 × 3

You can thus see that the sum of the first 10 terms follow this pattern

11× 13 × 1

11× 13 × 2

11× 13 × 3

11× 13 × 4

......

......

......

11× 13 × 4

11 × 13 × n

Just remember that n = 1 is the Fibonacci sequence starting with 1 and 1

n = 2 is the one starting with 2 and 2

And so forth....

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