Fibonacci sequence

The Fibonacci sequence is a famous mathematical sequence in which the first two terms are 1 and 1 and then each term after that is found by adding the previous two terms. The first 10 terms are shown in the figure below:

Mathematicians today are still finding interesting way this series of numbers can describe nature. To see how this sequence describes nature, take a close look at the figure above.

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

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 = F_6-1 + F_6-2 = F₅ + F₄ = 5 + 3 = 8

F₇ = F_7-1 + F_7-2 = 8 + 5 = 13

......

......

......

F_n = F_n-1 + F_n-2

How to 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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