3. For each of the sequences in parts (a), (b), and (c), write a Python function that accepts a positive integer n as input and returns an unrounded decimal approximation of the sequence’s nth term

Question

Use these functions to print the terms of each respective sequence for 2 ≤ n ≤ 15.

(a) The sequence of partial continued fractions is defined as

a_(1)=1, a_(2)=1+(1)/(1), a_(3)=1+(1)/(1+(1)/(1)), a_(4)=1+(1)/(1+(1)/(1+(1)/(1))),...

where the nth term of the sequence is defined recursively by an = 1 + 1 /a_n−1 for n ≥ 2.

(b) The sequence of nested roots is defined as

b_(1)=\sqrt(1),b_(2)=\sqrt(1+\sqrt(1)), b_(3)=\sqrt(1+\sqrt(1+\sqrt(1))),b_(4)=\sqrt(1+\sqrt(1+\sqrt(1+\sqrt(1)))),..

where the nth term of the sequence is defined recursively by b_(n)=\sqrt(1+b_(n)-1) for n ≥ 2.

(c) Recall that the Fibonacci numbers are defined recursively by

f_(n)=f_(n-1)+f_(n-2) 2 for n ≥ 3, where F1 = F2 = 1.

The sequence of Fibonacci ratios is defined as cn = Fn/Fn−1 for n ≥ 2.

(d) Plot the sequences an, bn, and cn for 2 ≤ n ≤ 15 in the same graph (see the overview for an example of plotting sequences). Next, compute and print an unrounded decimal approximation of φ = 1+√5/2. What do you notice about this value in relation to the sequences a_n, b_n, and c_n? Write a print statement describing your observations.

Answer 1

1. **Imports**

- `import math` allows the use of mathematical functions like `sqrt`.

- `import matplotlib.pyplot as plt` enables plotting with Matplotlib.

2. **Sequence Definitions**

1. **$(a(n))$**:

- Defines the **continued fraction** sequence iteratively:

$[a_1 = 1, \quad a_n = 1 + \frac{1}{a_{n-1}}]$

- It starts with `prev = 1.0` for $(a_1)$. For each subsequent term, it updates `prev` as `1 + 1 / prev`.

2. **$(b(n))$**:

- Defines the **nested square root** sequence:

$[b_1 = \sqrt{1}, \quad b_n = \sqrt{1 + b_{n-1}}]$

- It starts with `prev = 1.0` (i.e., $(\sqrt{1})$). For each term, it does `prev = math.sqrt(1 + prev)`.

3. **`fibonacci(n)`**:

- An **iterative** approach to compute the $(n)$-th Fibonacci number.

- It uses two variables, `a_val` and `b_val`, which it updates in a loop:

$[(a\_val, b\_val) \leftarrow (b\_val,\; a\_val + b\_val)]$

effectively generating Fibonacci numbers without recursion.

4. **$(c(n))$**:

- Computes the ratio of consecutive Fibonacci numbers, $( c(n) = F(n) / F(n-1) )$.

- It uses `fibonacci(n)` and `fibonacci(n-1)` to calculate the ratio.

3. **`main()` Function**

1. **Printing the Sequences**

- For $(n = 2)$ to $(15)$, prints the values of $(a(n))$, $(b(n))$, and $(c(n))$. This demonstrates how each sequence behaves over that range.

2. **Plotting the Sequences**

- Creates `n_values` from 2 to 15 and calculates corresponding $(a(n))$, $(b(n))$, and $(c(n))$.

- Plots all three sets of sequence values on the same graph (with distinct markers).

- Draws a **horizontal line** for the golden ratio $(\phi)$.

- Configures labels, legend, and grid for clarity.

3. **Golden Ratio and Observation**

- Calculates $(\phi = \frac{1 + \sqrt{5}}{2})$.

- Prints $(\phi)$ and notes that all three sequences converge to this value.

4. **Execution Guard**

- The `if __name__ == "__main__": main()` block ensures that `main()` is called only when the file is run directly, not when it’s imported as a module.

# Part (a): Continued fraction sequence

def a(n):

"""

Computes the nth term (for n >= 1) of the continued fraction sequence:

a_1 = 1,

a_n = 1 + 1 / a_(n-1), for n >= 2

Iteratively builds from n=1 to n.

"""

if n < 1:

raise ValueError("n must be a positive integer")

prev = 1.0 # This corresponds to a_1

# If n=1, we just return 1.0

for _ in range(2, n + 1):

prev = 1 + 1 / prev

return prev