The image you shared contains exercises involving the Fibonacci sequence. Let's address each question step by step.

1. Find Fib(8)

The Fibonacci sequence is defined as:

• Fib(1) = 1
• Fib(2) = 1
• Fib(3) = 2
• Fib(4) = 3
• Fib(5) = 5
• Fib(6) = 8
• Fib(7) = 13
• Fib(8) = 21

So, Fib(8) = 21.

2. Find Fib(19)

To find Fib(19), let's calculate further Fibonacci numbers:

• Fib(9) = 34
• Fib(10) = 55
• Fib(11) = 89
• Fib(12) = 144
• Fib(13) = 233
• Fib(14) = 377
• Fib(15) = 610
• Fib(16) = 987
• Fib(17) = 1597
• Fib(18) = 2584
• Fib(19) = 4181

Thus, Fib(19) = 4181.

3. If Fib(22) = 17,711 and Fib(24) = 46,368, what is Fib(23)?

To find Fib(23), we use the property of the Fibonacci sequence, which is:

• Fib(n) = Fib(n-1) + Fib(n-2)

We know:

• Fib(22) = 17,711
• Fib(24) = 46,368

So, we can calculate Fib(23) as: $\text{Fib(23)} = \text{Fib(24)} - \text{Fib(22)} = 46,368 - 17,711 = 28,657$

Thus, Fib(23) = 28,657.

Would you like further details or explanations on any of these steps?

Further Questions:

1. How can you derive the general formula for the Fibonacci sequence?
2. How do you calculate larger Fibonacci numbers using a recursive function in Python?
3. What is the significance of the Fibonacci sequence in nature and mathematics?
4. Can you prove the Fibonacci sequence through induction?
5. How does the Golden Ratio relate to the Fibonacci sequence?

Tip: When working with Fibonacci numbers, it can be useful to recognize patterns in their growth and even use matrix exponentiation for efficient calculation of very large Fibonacci numbers!