Let's tackle the questions one by one based on the information given.

Question 8: Find $F_7, F_8, F_9, F_{10}$, and $F_{11}$.

We know that the Fibonacci sequence is defined as: $F_n = F_{n-1} + F_{n-2}$ Starting from the given values:

• $F_1 = 1$
• $F_2 = 1$
• $F_3 = 2$
• $F_4 = 3$
• $F_5 = 5$
• $F_6 = 8$

Let's continue the sequence:

1. For $F_7$: $F_7 = F_6 + F_5 = 8 + 5 = 13$

2. For $F_8$: $F_8 = F_7 + F_6 = 13 + 8 = 21$

3. For $F_9$: $F_9 = F_8 + F_7 = 21 + 13 = 34$

4. For $F_{10}$: $F_{10} = F_9 + F_8 = 34 + 21 = 55$

5. For $F_{11}$: $F_{11} = F_{10} + F_9 = 55 + 34 = 89$

Therefore, the terms are: $F_7 = 13, \quad F_8 = 21, \quad F_9 = 34, \quad F_{10} = 55, \quad F_{11} = 89$

Question 9: For all $n \in \mathbb{Z}^+$, prove that $F_1 + F_3 + F_5 + \dots + F_{2n-1} = F_{2n}$.

This statement implies that the sum of the odd-indexed Fibonacci numbers up to $F_{2n-1}$ is equal to $F_{2n}$.

Proof by Induction:

1. Base Case: $n = 1$
   $F_1 = F_2 = 1$
   So, the base case holds.

2. Inductive Step:
   Assume $F_1 + F_3 + F_5 + \dots + F_{2k-1} = F_{2k}$ holds.
   We need to prove that $F_1 + F_3 + F_5 + \dots + F_{2k-1} + F_{2k+1} = F_{2k+2}$.

   Since $F_{2k+2} = F_{2k+1} + F_{2k}$ and by our inductive hypothesis $F_1 + F_3 + \dots + F_{2k-1} = F_{2k}$, we have: $F_1 + F_3 + \dots + F_{2k+1} = F_{2k} + F_{2k+1} = F_{2k+2}$ Thus, the statement holds by induction.

Question 10: For all $n \in \mathbb{Z}^+$, prove that $F_1^2 + F_2^2 + F_3^2 + \dots + F_n^2 = F_n F_{n+1}$.

This states that the sum of the squares of the first $n$ Fibonacci numbers is equal to the product of $F_n$ and $F_{n+1}$.

Proof by Induction:

1. Base Case: $n = 1$
   $F_1^2 = 1 \quad \text{and} \quad F_1 F_2 = 1 \times 1 = 1$
   So, the base case holds.

2. Inductive Step:
   Assume $F_1^2 + F_2^2 + \dots + F_k^2 = F_k F_{k+1}$ holds for $n = k$.
   We need to prove $F_1^2 + F_2^2 + \dots + F_k^2 + F_{k+1}^2 = F_{k+1} F_{k+2}$.

   From the inductive hypothesis, $F_1^2 + F_2^2 + \dots + F_k^2 = F_k F_{k+1}$.
   Adding $F_{k+1}^2$ to both sides, we get: $F_1^2 + F_2^2 + \dots + F_k^2 + F_{k+1}^2 = F_k F_{k+1} + F_{k+1}^2$ $= F_{k+1}(F_k + F_{k+1}) = F_{k+1} F_{k+2}$ Therefore, the statement holds by induction.

Would you like further details on any of these proofs or have any questions?

Related Questions:

1. What is the general formula for the Fibonacci sequence?
2. How can we express Fibonacci numbers using matrix exponentiation?
3. Can the sum of even-indexed Fibonacci numbers be represented similarly?
4. What is the closed-form formula for Fibonacci numbers known as Binet's formula?
5. How can we prove that every nth Fibonacci number is divisible by $F_n$?

Tip:

To explore Fibonacci numbers further, try deriving their properties using Binet's formula. It provides an exact form and can be helpful in proofs involving Fibonacci identities.