Storing answers to sub-problems is an essential aspect of dynamic programming. Let’s explore why this is the case.

The Fibonacci Sequence is a series of numbers where the next number equals the sum of the previous two, starting with `0`

and `1`

.

Here’s a list of the first 10 Fibonacci numbers: `0, 1, 1, 2, 3, 5, 8, 13, 21`

. The zeroth Fibonacci number is 0.

We can express this sequence as a function:

f(n) = f(n - 1) + f(n - 2)

We’ve written a Python function that returns the `n`

th Fibonacci number.

Let’s add some print statements so we can see how often we’re repeating work in the current implementation.

### Instructions

**1.**

We need a way to store our function calls.

Above the `fibonacci()`

function, declare `arguments_count`

as an empty dictionary.

`arguments_count`

will hold the argument as a key and the number of times we called `fibonacci()`

with that argument as the value.

**2.**

We need to fill our dictionary so we can log the results.

Inside the first line of the `fibonacci()`

function, declare the variable `count`

. Set `count`

to be the value located in `arguments_count`

under the key `num`

.

**Sometimes that key won’t exist because the argument hasn’t been seen before.**

Use `0`

as a default. What dictionary method would be useful here?

**3.**

With a sensible default in place, reassign `count`

to be one more than it was previously to mark **this** function call.

After incrementing, store the updated value back in `arguments_count`

with the same key of `num`

.

**4.**

We’ve successfully completed our logging. Run the function.

See how often arguments are repeated? Change `num_in_fibonacci`

to different numbers.

The bigger the number, the more repetitions!

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