Fibonacci sequence
The Fibonacci sequence is a naturally occuring 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 decribes nature, take a close look at the figure below:
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
F
_{1} = 1
F
_{2} = 1
F
_{3} = F
_{2} + F
_{1} = 1 + 1 = 2
F
_{4} = F
_{3} + F
_{2} = 2 + 1 = 3
F
_{5} = F
_{4} + F
_{3} = 3 + 2 = 5
F
_{6} = F
_{61} + F
_{62} = F
_{5} + F
_{4} = 5 + 3 = 8
F
_{7} = F
_{71} + F
_{72} = 8 + 5 = 13
......
......
......
F
_{n} = F
_{n1} + F
_{n2}
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 Fibonaccilike 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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