Let's break down and answer each part of the problem on the image you uploaded, focusing on the Fibonacci sequence. Here is a solution step-by-step.

Fibonacci Sequence:

The Fibonacci sequence is defined as:

• Fib(1) = 1
• Fib(2) = 1
• Fib(n) = Fib(n-1) + Fib(n-2) for n > 2.

1. Find Fib(8)

Fib(8) = Fib(7) + Fib(6)

• Fib(7) = Fib(6) + Fib(5)
• Fib(6) = Fib(5) + Fib(4)
• Fib(5) = 5 (from the known sequence)

Calculation:

• Fib(8) = 21

2. Find Fib(19)

• Through continued calculation of the sequence or using a formula: Fib(19) = 4181.

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

Since Fib(23) = Fib(24) - Fib(22),

• Fib(23) = 46,368 - 17,711 = 28,657.

4. Evaluate the sums:

a. Fib(1) + Fib(2)

• Fib(1) + Fib(2) = 1 + 1 = 2

b. Fib(1) + Fib(2) + Fib(3)

• Fib(3) = 2, so:
• 1 + 1 + 2 = 4

c. Fib(1) + Fib(2) + Fib(3) + Fib(4)

• Fib(4) = 3, so:
• 1 + 1 + 2 + 3 = 7

5. Determine the pattern and sum up to Fib(10)

Pattern: Each sum is the next Fibonacci number minus 1. So, for Fib(10):

• Fib(10) = 55.
• The sum is Fib(10 + 2) - 1 = Fib(12) - 1 = 144 - 1 = 143.

6. Evaluate the products:

a. Fib(3) x Fib(5)

• Fib(3) = 2, Fib(5) = 5, so:
• 2 × 5 = 10.

b. Fib(4) x Fib(1)

• Fib(4) = 3, Fib(1) = 1, so:
• 3 × 1 = 3.

c. Fib(3) x Fib(5) + Fib(2)

• 10 + Fib(2) = 10 + 1 = 11.

7. Find Fib(15)

Fib(15) = 610.

8. Find Fib(20)

Fib(20) = 6765.

9. If Fib(30) = 832,040 and Fib(28) = 317,811, what is Fib(29)?

Fib(29) = Fib(30) - Fib(28) = 832,040 - 317,811 = 514,229.

10. What is Fib(8)?

We found earlier that Fib(8) = 21.

Do you want further details or clarifications on any step? Here are a few related questions:

1. How does the Fibonacci formula work for large numbers?
2. Can you find a general pattern in the Fibonacci sums?
3. What is the importance of Fibonacci numbers in nature?
4. What is the relationship between Fibonacci numbers and the golden ratio?
5. How can Fibonacci sequences be applied in computer algorithms?

Tip: Fibonacci numbers can be quickly calculated using the Binet formula for large values!