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The question you're working on involves proving a matrix relation for Fibonacci numbers using mathematical induction. Let's break down the process.

The conjecture states that:

\[
\begin{pmatrix}
1 & 1 \\
1 & 0
\end{pmatrix}^n
=
\begin{pmatrix}
F_{n+1} & F_n \\
F_n & F_{n-1}
\end{pmatrix}
\]

where \( F_n \) represents the Fibonacci numbers. To prove this by induction, we'll follow these steps:

### Step 1: Base Case \( n = 1 \)
For \( n = 1 \), we check whether the conjecture holds true:

\[
\begin{pmatrix}
1 & 1 \\
1 & 0
\end{pmatrix}^1
=
\begin{pmatrix}
1 & 1 \\
1 & 0
\end{pmatrix}
\]

We need to confirm that this equals:

\[
\begin{pmatrix}
F_2 & F_1 \\
F_1 & F_0
\end{pmatrix}
=
\begin{pmatrix}
1 & 1 \\
1 & 0
\end{pmatrix}
\]

which is clearly true, as \( F_2 = 1 \), \( F_1 = 1 \), and \( F_0 = 0 \). So, the base case holds.

### Step 2: Inductive Hypothesis
Assume that the conjecture holds for some arbitrary \( n = k \), i.e.,

\[
\begin{pmatrix}
1 & 1 \\
1 & 0
\end{pmatrix}^k
=
\begin{pmatrix}
F_{k+1} & F_k \\
F_k & F_{k-1}
\end{pmatrix}
\]

### Step 3: Inductive Step
We need to prove that the conjecture holds for \( n = k+1 \). This means we must show that:

\[
\begin{pmatrix}
1 & 1 \\
1 & 0
\end{pmatrix}^{k+1}
=
\begin{pmatrix}
F_{k+2} & F_{k+1} \\
F_{k+1} & F_k
\end{pmatrix}
\]

Using matrix multiplication:

\[
\begin{pmatrix}
1 & 1 \\
1 & 0
\end{pmatrix}^{k+1}
=
\begin{pmatrix}
1 & 1 \\
1 & 0
\end{pmatrix}
\begin{pmatrix}
F_{k+1} & F_k \\
F_k & F_{k-1}
\end{pmatrix}
\]

Carry out the multiplication:

\[
\begin{pmatrix}
1 & 1 \\
1 & 0
\end{pmatrix}
\begin{pmatrix}
F_{k+1} & F_k \\
F_k & F_{k-1}
\end{pmatrix}
=
\begin{pmatrix}
F_{k+1} + F_k & F_k + F_{k-1} \\
F_{k+1} & F_k
\end{pmatrix}
\]

Using the Fibonacci recurrence relation \( F_{n+1} = F_n + F_{n-1} \), this simplifies to:

\[
\begin{pmatrix}
F_{k+2} & F_{k+1} \\
F_{k+1} & F_k
\end{pmatrix}
\]

which is exactly what we needed to show.

### Conclusion:
By induction, the conjecture holds for all \( n \geq 1 \).

---

Do you want more details or have any questions? Here are 5 related questions:

1. How are Fibonacci numbers defined recursively?
2. Can you apply this matrix form of Fibonacci numbers to compute large Fibonacci numbers more efficiently?
3. What other matrix identities are useful in number theory?
4. How does the Fibonacci sequence relate to the golden ratio?
5. How can this matrix form be used in computer algorithms for fast Fibonacci computation?

**Tip:** Matrix exponentiation is a highly efficient method to compute Fibonacci numbers, especially for large indices. It reduces the time complexity to \( O(\log n) \).

Assume the conjecture is true for n = n and test the validity for n = n + 1. Write down your conclusion.

Explore the matrix representation of Fibonacci numbers and how to prove it using mathematical induction. Learn the conjecture involving the matrix (1 1; 1 0) raised to the power n and its relation to Fibonacci numbers.

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