import math

def fibonacci_closed_form(n):

phi = (1 + math.sqrt(5)) / 2

return round((phi**n - (-1/phi)**n) / math.sqrt(5))

# Define the number of Fibonacci numbers you want to print

num_terms = 5

# Print the Fibonacci sequence

for i in range(num_terms):

print(fibonacci_closed_form(i))

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### Explanation:

The import math statement in Python allows you to access various mathematical functions and constants provided by the Python math module. These functions include trigonometric functions, logarithmic functions, and mathematical constants such as ฯ (pi) and e.

In the given code snippet, import math is used to access the sqrt() function from the math module. This function calculates the square root of a number.

The fibonacci_closed_form() function uses the golden ratio (phi) and its inverse to compute the nth Fibonacci number using Binet's formula. Binet's formula is a closed-form expression that directly calculates the nth Fibonacci number without recursion or iteration.

Here's a breakdown of the fibonacci_closed_form() function:

It calculates the golden ratio (phi) using the formula (1 + math.sqrt(5)) / 2.

Then, it uses phi and its inverse (-1/phi) to compute the nth Fibonacci number using Binet's formula:

Finally, it rounds the result using the round() function to get the nearest integer, which is the nth Fibonacci number.

The code then prints the Fibonacci sequence for a specified number of terms (num_terms) using a loop. Each iteration of the loop calls the fibonacci_closed_form() function to compute the nth Fibonacci number and prints the result.

def fibonacci_generator():

a, b = 0, 1

while True:

yield a

a, b = b, a + b

sequence = fibonacci_generator()

for _ in range(6):

print(next(sequence))

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### Explanantion:

Let's break down the code:

def fibonacci_generator():

a, b = 0, 1

while True:

yield a

a, b = b, a + b

This defines a generator function fibonacci_generator(). Generators are special functions in Python that allow you to generate a sequence of values lazily, i.e., one at a time, rather than generating them all at once and storing them in memory.

Inside the function, two variables a and b are initialized to 0 and 1 respectively. These variables are used to generate the Fibonacci sequence.

The while True loop creates an infinite loop, which means the generator will continue generating Fibonacci numbers indefinitely.

Inside the loop, the yield keyword is used to yield (or produce) the current value of a, making this function a generator. yield pauses the function's execution and returns a value to the caller without exiting the function.

After yielding a, the values of a and b are updated for the next iteration of the loop. This simulates the Fibonacci sequence generation logic where each number is the sum of the two preceding numbers.

sequence = fibonacci_generator()

for _ in range(6):

print(next(sequence))

Here, fibonacci_generator() is called to create a generator object named sequence. This generator object will produce Fibonacci numbers when iterated over.

Then, a loop runs six times, each time calling next(sequence). This retrieves the next value from the generator sequence. The _ variable is used as a placeholder for the loop variable, as its value is not used inside the loop. The print() function then prints each Fibonacci number generated by the generator function.

So, the output of this code will be the first six Fibonacci numbers:

0

1

1

2

3

5

And the generator will continue to generate Fibonacci numbers if you keep calling next(sequence).

def fibonacci_memoization(n, memo={}):

if n in memo:

return memo[n]

if n <= 1:

return n

else:

memo[n] = fibonacci_memoization(n-1, memo) + fibonacci_memoization(n-2, memo)

return memo[n]

# Define the number of Fibonacci numbers you want to print

num_terms = 6

# Print the Fibonacci sequence

for i in range(num_terms):

print(fibonacci_memoization(i))

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### Explanation:

This code implements the Fibonacci sequence using memoization to optimize performance by avoiding redundant calculations. Let's break down how it works:

def fibonacci_memoization(n, memo={}):

if n in memo:

return memo[n]

if n <= 1:

return n

else:

memo[n] = fibonacci_memoization(n-1, memo) + fibonacci_memoization(n-2, memo)

return memo[n]

The function fibonacci_memoization takes two parameters: n, which is the index of the Fibonacci number to compute, and memo, which is a dictionary used to store previously computed Fibonacci numbers to avoid redundant calculations.

It first checks if the Fibonacci number for index n is already in the memo dictionary. If it is, it returns the value directly from the memo, avoiding recalculation.

If the Fibonacci number for index n is not in the memo, it proceeds to compute it.

If n is 0 or 1, it returns n directly since the Fibonacci sequence starts with 0 and 1.

Otherwise, it recursively computes the (n-1)th and (n-2)th Fibonacci numbers using memoization, adds them together, stores the result in the memo dictionary, and returns the result.

This memoization technique ensures that each Fibonacci number is computed only once, significantly reducing the number of recursive calls needed to generate the sequence.

# Define the number of Fibonacci numbers you want to print

num_terms = 6

# Print the Fibonacci sequence

for i in range(num_terms):

print(fibonacci_memoization(i))

Here, num_terms specifies the number of Fibonacci numbers to print.

The for loop iterates num_terms times, computing and printing each Fibonacci number using the fibonacci_memoization function.

When you run this code, it will print the first six Fibonacci numbers:

0

1

1

2

3

5

And it will continue generating Fibonacci numbers indefinitely if you continue calling fibonacci_memoization(i).

def fibonacci_recursive(n):

if n <= 1:

return n

else:

return fibonacci_recursive(n-1) + fibonacci_recursive(n-2)

# Define the number of Fibonacci numbers you want to print

num_terms = 5

# Print the Fibonacci sequence

for i in range(num_terms):

print(fibonacci_recursive(i))

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### Explanation:

This code defines a recursive function fibonacci_recursive to generate the Fibonacci sequence. Let's break it down:

def fibonacci_recursive(n):

if n <= 1:

return n

else:

return fibonacci_recursive(n-1) + fibonacci_recursive(n-2)

The function fibonacci_recursive takes a single argument n, which represents the index of the Fibonacci number to compute.

If n is less than or equal to 1, meaning it's either 0 or 1, the function returns n itself. This is because the Fibonacci sequence starts with 0 and 1.

If n is greater than 1, the function recursively computes the (n-1)th and (n-2)th Fibonacci numbers by calling itself with n-1 and n-2, respectively, and then adds them together to get the nth Fibonacci number.

# Define the number of Fibonacci numbers you want to print

num_terms = 5

# Print the Fibonacci sequence

for i in range(num_terms):

print(fibonacci_recursive(i))

Here, num_terms specifies the number of Fibonacci numbers to print.

The for loop iterates num_terms times, starting from 0 and ending at num_terms - 1. In each iteration, it computes and prints the Fibonacci number for the current index using the fibonacci_recursive function.

When you run this code, it will print the first five Fibonacci numbers:

0

1

1

2

3

And it will continue generating Fibonacci numbers indefinitely if you continue calling fibonacci_recursive(i). However, recursive Fibonacci calculation can become inefficient for large values of n due to redundant computations.

def fibonacci_iterative(n):

fib_sequence = [0, 1]

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

fib_sequence.append(fib_sequence[-1] + fib_sequence[-2])

return fib_sequence

sequence = fibonacci_iterative(5)

print(sequence)

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### Explanation:

This code defines a function fibonacci_iterative to generate the Fibonacci sequence using an iterative approach. Let's go through it step by step:

def fibonacci_iterative(n):

fib_sequence = [0, 1]

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

fib_sequence.append(fib_sequence[-1] + fib_sequence[-2])

return fib_sequence

The function fibonacci_iterative takes a single argument n, which represents the number of Fibonacci numbers to generate.

It initializes a list fib_sequence with the first two Fibonacci numbers, 0 and 1.

It then enters a loop starting from index 2 (since the first two Fibonacci numbers are already in the list) up to n. In each iteration, it calculates the next Fibonacci number by adding the last two numbers in the sequence (fib_sequence[-1] and fib_sequence[-2]) and appends it to the list.

After the loop completes, the function returns the entire list fib_sequence containing the generated Fibonacci sequence.

sequence = fibonacci_iterative(5)

print(sequence)

This code calls the fibonacci_iterative function with n = 5 to generate the first 5 Fibonacci numbers.

It then prints the generated sequence.

When you run this code, it will print the first 5 Fibonacci numbers:

[0, 1, 1, 2, 3, 5]

This iterative approach is efficient and straightforward for generating Fibonacci numbers, especially for small values of n.

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