(d) Write the following functions: i. fib( n ) that returns the nth term of a Fibonacci, $$ \mathrm{f}(\mathrm{n}) $$, sequence, given that: $$ f(1)=1 ; f(2)=1 ; f(n)=f(n-\mathbf{1})+f(n-2), n \geq 3 $$ ii. fibSum ( n ) that returns the sum of the first n terms of a Fibonacci sequence. Utilize the previous function. iii. fibRatio ( n ) that returns the ratio $$ \mathrm{f}(\mathrm{n}+1) \mathrm{f}(\mathrm{n}), \mathrm{n} \geq 1 $$. Utilize the first function. [ 2 marks] iv. Assuming that the three functions are in one program saved as mod. py in a folder called mods and you create a program in the same folder as the folder. Demonstrate how you would import the package into your program. Print the following in the program. Do it twice by first using importing the package and secondly by importing the module inside the package (as m). The tenth term of the Fibonacci sequence. The sum of the first 10 terms of the sequence and the 11 th term divided by the 10th term. The printed message should be: $$ [6 $$ marks $$ ] $$ The 10 th term is 55, the sum is 143 and the ratio is 1.6181818181818182 .

Question

UpStudy AI · Accepted Answer

Certainly! Below are the implementations for each part of your problem, including the functions `fib`, `fibSum`, `fibRatio`, and the demonstration of importing the module in two different ways.

---

### **i. `fib(n)` Function**
This function computes the nth term of the Fibonacci sequence using recursion.

```python
# mod.py

def fib(n):
    if n == 1 or n == 2:
        return 1
    else:
        return fib(n - 1) + fib(n - 2)
```

---

### **ii. `fibSum(n)` Function**
This function calculates the sum of the first `n` terms of the Fibonacci sequence by utilizing the `fib` function.

```python
# mod.py

def fibSum(n):
    total = 0
    for i in range(1, n + 1):
        total += fib(i)
    return total
```

---

### **iii. `fibRatio(n)` Function**
This function computes the ratio $$ \frac{f(n+1)}{f(n)} $$ using the `fib` function.

```python
# mod.py

def fibRatio(n):
    return fib(n + 1) / fib(n)
```

---

### **iv. Importing the Module and Using the Functions**

Assuming the three functions are saved in a file named `mod.py` inside a folder called `mods`, and you have another Python script in the same parent directory, here's how you can import and use the functions in two different ways.

**Folder Structure:**
```
your_project/
│
├── mods/
│   ├── __init__.py
│   └── mod.py
│
└── main.py
```

**Contents of `mod.py`:**
```python
# mods/mod.py

def fib(n):
    if n == 1 or n == 2:
        return 1
    else:
        return fib(n - 1) + fib(n - 2)

def fibSum(n):
    total = 0
    for i in range(1, n + 1):
        total += fib(i)
    return total

def fibRatio(n):
    return fib(n + 1) / fib(n)
```

**Ensure `__init__.py` Exists:**
To make `mods` a package, ensure there's an `__init__.py` file inside the `mods` folder. It can be empty or can contain initialization code.

```python
# mods/__init__.py

# This file can be left empty
```

**Importing and Using the Module in `main.py`:**

```python
# main.py

# First Method: Importing the entire module through the package
import mods.mod

# Using the functions with full module path
term_10 = mods.mod.fib(10)
sum_10 = mods.mod.fibSum(10)
ratio_10 = mods.mod.fibRatio(10)
print(f"The 10th term is {term_10}, the sum is {sum_10} and the ratio is {ratio_10}.")

# Second Method: Importing the module with an alias
from mods import mod as m

# Using the functions with the alias
term_10_m = m.fib(10)
sum_10_m = m.fibSum(10)
ratio_10_m = m.fibRatio(10)
print(f"The 10th term is {term_10_m}, the sum is {sum_10_m} and the ratio is {ratio_10_m}.")
```

**Output When Running `main.py`:**
```
The 10th term is 55, the sum is 143 and the ratio is 1.6181818181818182.
The 10th term is 55, the sum is 143 and the ratio is 1.6181818181818182.
```

---

### **Explanation:**

1. **Function Implementations:**
   - `fib(n)`: Recursively calculates the nth Fibonacci number.
   - `fibSum(n)`: Iteratively sums up the first `n` Fibonacci numbers by calling `fib(i)` for each term.
   - `fibRatio(n)`: Computes the ratio of the (n+1)th term to the nth term.

2. **Import Methods:**
   - **First Method (`import mods.mod`):** This imports the `mod` module from the `mods` package. Functions are accessed using the full path `mods.mod.function_name`.
   - **Second Method (`from mods import mod as m`):** This imports the `mod` module with an alias `m`, allowing functions to be accessed as `m.function_name`.

3. **Printing the Results:**
   - Both methods are used to print the same results, demonstrating that both import techniques are effective and produce identical outputs.

This setup ensures modularity and reusability of your Fibonacci functions across different programs within the same project.
```python # mod.py def fib(n): if n == 1 or n == 2: return 1 else: return fib(n - 1) + fib(n - 2) def fibSum(n): total = 0 for i in range(1, n + 1): total += fib(i) return total def fibRatio(n): return fib(n + 1) / fib(n) ``` **main.py:** ```python # main.py # First Method: Importing the entire module through the package import mods.mod # Using the functions with full module path term_10 = mods.mod.fib(10) sum_10 = mods.mod.fibSum(10) ratio_10 = mods.mod.fibRatio(10) print(f"The 10th term is {term_10}, the sum is {sum_10} and the ratio is {ratio_10}.") # Second Method: Importing the module with an alias from mods import mod as m # Using the functions with the alias term_10_m = m.fib(10) sum_10_m = m.fibSum(10) ratio_10_m = m.fibRatio(10) print(f"The 10th term is {term_10_m}, the sum is {sum_10_m} and the ratio is {ratio_10_m}.") ``` **Output:** ``` The 10th term is 55, the sum is 143 and the ratio is 1.6181818181818182. The 10th term is 55, the sum is 143 and the ratio is 1.6181818181818182. ```

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