# An alternate way to calculate Fibonacci series

###
Muhammed H. Alkan
*Updated on *
・2 min read

###### (The alternate way is at the end!)

Fibonacci series is a collection of numbers which two numbers that precede it.

Which will produce a series like this:

`0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377`

## Calculating it

There are two popular ways to do this

- Using the normal method (
`1 + 1 = 2`

,`1 + 2 = 3`

,`3 + 2 = 5`

,`...`

) - Using the recursive method (
`Fn = F(n - 1) + F(n - 2)`

where`F1`

and`F2`

is equal to`1`

)

Implementations in Python will be something like this (Finding first 100 Fibonacci numbers)

- Normal method

```
a, b = 0, 1
for _ in range(100)
print(a, end=' ')
a, b = b, a + b
#=> 0 1 1 2 3 5 8 13...
```

- Recursive method

```
def fib(n):
if n < 3:
return 1
return fib(n - 1) + fib(n - 2)
for n in range(100):
print(fib(n), end = ' ')
#=> 0 1 1 2 3 5 8 13...
```

###### (Not recommended to do it in Python due Tail Call Optimization)

So until now, I haven't seen any other ways. Why not find another way?

There is an interesting fact in Fibonacci numbers. What is the fact?

-> Dividing `F(n + 1)`

with `Fn`

(`F(n + 1) / Fn`

) will approximate the golden ratio! (`1.61803399...`

)

Let us test it!

`1, 1, 2, 3, 5, 8, 13, 21, 34...`

```
1 / 1 = 1
2 / 1 = 2
3 / 2 = 1.5
8 / 5 = 1.6
13 / 8 = 1.62
21 / 13 = 1.615
34 / 21 = 1.619
...
```

And so on! While numbers getting numbers bigger, the ratio will be more similar to the golden ratio. Then, what we can do with it?

If `F(n + 1) / Fn ≈ φ`

, `Fn * φ`

will be (approximately) equal to `F(n + 1)`

which founds the next fibonacci number!

So, let's implement it in Python!

```
fib = 1
for _ in range(100):
print(fib, end=' ')
fib = round(fib * 1.618)
#=> 1, 1, 2, 3, 5, 8, 13, 21, 34...`
```

So I multiplied `fib`

with `1.618`

which is approximate golden ratio. As I said it will be found next fibonacci number approximately. Then I rounded it because I want an integer, not a decimal. If we need to visualize it:

```
fib = 1
fib = round(fib * 1.61) -> fib = round(1.61) -> fib = 2
fib = round(fib * 1.61) -> fib = round(3.22) -> fib = 3
fib = round(fib * 1.61) -> fib = round(4.83) -> fib = 5
fib = round(fib * 1.61) -> fib = round(8.05) -> fib = 8
...
```

And so on! It's an interesting method (found by me :p), but it can be wrong for really big numbers. For big numbers, you will need a better approximation of the golden ratio to use it. Because the decimal numbers multiplying with big numbers will create bigger numbers. For example:

```
1.2345 * 10 = 12.345
1.2345 * 100 = 123.45
1.2345 * 1000 = 1234.5
1.2345 * 10000 = 12345
1.2345 * 100000 = 123450
```

So basically I don't recommend this method for big numbers.

Thanks for reading this post!

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