A recursive formula can be implemented in Python and then used to provide the nth term in a series.
A recursive formula can be implemented in Python and then used to provide the nth term in a series.
Computer Networking: A Top-Down Approach (7th Edition)
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Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
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![### Question 15
A recursive formula can be implemented in Python and then used to provide the nth term in a series.
Which of the following recursive programs implements the recursive formula below and prints out its 5th term (-8):
\[
\begin{align*}
\text{base: } a(1) &= 12 \\
\text{recursion: } a(n) &= a(n - 1) - 5 \\
\end{align*}
\]
```python
def recursive_3(num):
if num == 1:
return 12
return recursive_3(num - 1) - 5
print('rec_3 ', recursive_3(5))
```
The provided code implements the given recursive formula. Here is a breakdown of the code:
- The function `recursive_3` is defined with a single parameter, `num`.
- **Base Case**: If `num` equals 1, the function returns 12.
- **Recursive Case**: If `num` is not 1, the function calls itself with the argument `num - 1` and then subtracts 5 from the result.
- Finally, the function is invoked with the argument 5, and the result is printed.
The code correctly implements the recursive formula and calculates the 5th term as follows:
\[
\begin{align*}
a(5) &= a(4) - 5 \\
a(4) &= a(3) - 5 \\
a(3) &= a(2) - 5 \\
a(2) &= a(1) - 5 \\
a(1) &= 12 \\
\end{align*}
\]
\[
\begin{align*}
a(2) &= 12 - 5 = 7 \\
a(3) &= 7 - 5 = 2 \\
a(4) &= 2 - 5 = -3 \\
a(5) &= -3 - 5 = -8 \\
\end{align*}
\]
Thus, the 5th term is correctly calculated to be -8.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa9ed15ea-b247-4d4a-84f5-d7f808a88737%2F150cd8d1-313d-473a-8244-9489b0aa41c7%2Fjk379om_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question 15
A recursive formula can be implemented in Python and then used to provide the nth term in a series.
Which of the following recursive programs implements the recursive formula below and prints out its 5th term (-8):
\[
\begin{align*}
\text{base: } a(1) &= 12 \\
\text{recursion: } a(n) &= a(n - 1) - 5 \\
\end{align*}
\]
```python
def recursive_3(num):
if num == 1:
return 12
return recursive_3(num - 1) - 5
print('rec_3 ', recursive_3(5))
```
The provided code implements the given recursive formula. Here is a breakdown of the code:
- The function `recursive_3` is defined with a single parameter, `num`.
- **Base Case**: If `num` equals 1, the function returns 12.
- **Recursive Case**: If `num` is not 1, the function calls itself with the argument `num - 1` and then subtracts 5 from the result.
- Finally, the function is invoked with the argument 5, and the result is printed.
The code correctly implements the recursive formula and calculates the 5th term as follows:
\[
\begin{align*}
a(5) &= a(4) - 5 \\
a(4) &= a(3) - 5 \\
a(3) &= a(2) - 5 \\
a(2) &= a(1) - 5 \\
a(1) &= 12 \\
\end{align*}
\]
\[
\begin{align*}
a(2) &= 12 - 5 = 7 \\
a(3) &= 7 - 5 = 2 \\
a(4) &= 2 - 5 = -3 \\
a(5) &= -3 - 5 = -8 \\
\end{align*}
\]
Thus, the 5th term is correctly calculated to be -8.
Transcribed Image Text:### Understanding Recursive Functions in Python
This educational segment will help you understand how recursive functions work in Python. Below are three examples of recursive functions, `recursive_3`, with slight variations in their implementation.
#### Example 1
```python
def recursive_3(num):
if num == 1:
return 12
return recursive_3(num + 1) - 5
print('rec_3', recursive_3(5))
```
**Explanation:**
1. **Base Case:** If `num` is 1, the function returns 12.
2. **Recursive Step:** If `num` is not 1, it calls itself with `num + 1` and subtracts 5 from the result.
This pattern creates a deeper recursive call each time until `num` eventually equals 1.
#### Example 2
```python
def recursive_3(num):
if num == 1:
return -5
return recursive_3(num - 1) + 12
print('rec_3', recursive_3(5))
```
**Explanation:**
1. **Base Case:** If `num` is 1, the function returns -5.
2. **Recursive Step:** If `num` is not 1, it calls itself with `num - 1` and adds 12 to the result.
This pattern reduces `num` each time until it equals 1.
#### Example 3
```python
def recursive_3(num):
if num == 12:
return 12
return recursive_3(num - 1) - 5
print('rec_3', recursive_3(5))
```
**Explanation:**
1. **Base Case:** If `num` is 12, the function returns 12.
2. **Recursive Step:** If `num` is not 12, it calls itself with `num - 1` and subtracts 5 from the result.
This pattern reduces `num` until it equals 12.
### Common Pattern
- **Base Case:** An essential part of any recursive function, it stops further recursive calls.
- **Recursive Step:** The function calls itself with a modified argument, edging closer to the base case.
### Graphical Representation
While not explicitly shown in the images, one could visualize these recursive functions with call stacks, where each function call is placed on the stack until the base case
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