## Table of Contents

## Understanding the Problem

The `DoublePennies()`

function takes two arguments:

`pennies`

: The initial number of pennies (an integer).`days`

: The number of days (an integer) for which the pennies will be doubled.

The function should return the total number of pennies after doubling them for the specified number of days.

## Implementing the Solution

To implement the solution, we'll use *recursion*. Recursion is a programming technique where a function calls itself to solve a problem.

### Base Case

The base case is the simplest form of the problem that can be solved directly. In our case, the base case occurs when there are no days left to double the pennies. In this case, we simply return the current number of pennies.

```
def DoublePennies(pennies, days):
if days == 0:
return pennies
```

### Recursive Case

For the recursive case, we'll call the `DoublePennies()`

function with the updated number of pennies (i.e., double the current number of pennies) and a decremented number of days (i.e., one less day to double the pennies).

```
def DoublePennies(pennies, days):
if days == 0:
return pennies
else:
return DoublePennies(pennies * 2, days - 1)
```

## Sample Output

Here's an example of the `DoublePennies()`

function in action:

```
print(DoublePennies(1, 5)) # Output: 32
```

In this example, we start with 1 penny and double it for 5 days. The function returns 32, which is the total number of pennies after 5 days of doubling.

## FAQs

### 1. Why use recursion instead of a loop?

Recursion can often make the code more elegant and easier to understand by breaking the problem down into smaller, more manageable subproblems. However, recursion can also be less efficient than a loop, as it requires additional memory for each function call. In some cases, a loop may be more appropriate, depending on the specific requirements and constraints of the problem.

### 2. What is the time complexity of the `DoublePennies()`

function?

The time complexity of the `DoublePennies()`

function is O(n), where n is the number of days. This is because the function makes one recursive call for each day.

### 3. Can the `DoublePennies()`

function be optimized?

Yes, the `DoublePennies()`

function can be optimized using a technique called *memoization*. Memoization involves storing the results of expensive function calls and returning the cached result when the same inputs occur again. This can significantly reduce the number of recursive calls and improve the function's performance.

### 4. What are some common issues with recursive functions?

Some common issues with recursive functions include:

- Incorrect base case: Failing to correctly specify the base case can result in an infinite loop or incorrect output.
- Stack overflow: Recursive functions can cause a stack overflow if the recursion depth becomes too large. This can happen if the function is called with a large number of days or if the recursion is not well-structured.

### 5. How can I test the `DoublePennies()`

function?

You can test the `DoublePennies()`

function by calling it with different input values and checking the output against the expected result. For example, you can create a series of test cases with known input-output pairs and use assertions to verify that the function produces the correct output for each test case.