Let's break down each part of the problem step by step:

Part 1: Fibonacci Sequence Calculation

We are given the standard Fibonacci sequence where:

• Fib(1) = 1
• Fib(2) = 1
• Fib(3) = 2
• Fib(4) = 3, and so on, where each term is the sum of the two preceding terms.

a) Fib(8):

Using the Fibonacci rule $\text{Fib}(n) = \text{Fib}(n-1) + \text{Fib}(n-2)$:

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

b) Fib(15):

Continuing the pattern:

• Fib(9) = 34
• Fib(10) = 55
• Fib(11) = 89
• Fib(12) = 144
• Fib(13) = 233
• Fib(14) = 377
• Fib(15) = 610

c) Fib(20):

You stated that Fib(20) = 790. Hence, this is provided directly.

Part 2: Finding Fib(21) and Fib(19) using Fib(20) = 790 and Fib(22) = 920

a) Fib(21):

We know from the Fibonacci property that: $\text{Fib}(22) = \text{Fib}(21) + \text{Fib}(20)$ Thus, Fib(21) can be found as: $920 = \text{Fib}(21) + 790$ $\text{Fib}(21) = 920 - 790 = 130$

b) Fib(19):

Using the same idea: $\text{Fib}(20) = \text{Fib}(19) + \text{Fib}(18)$ But we can also use the reverse property: $\text{Fib}(21) = \text{Fib}(20) + \text{Fib}(19)$ Substituting the values: [ 130 = 790 +