The Fibonacci numbers are defined by the recurrence relation:

$F_n = F_{n-1} + F_{n-2}$ with $F_0 = 0$ and $F_1 = 1$.

The Binet formula for the $n$-th Fibonacci number is:

$F_n = \frac{1}{\sqrt{5}} \left( \phi^n - \hat{\phi}^n \right)$

where:

• $\phi = \frac{1 + \sqrt{5}}{2}$ (the golden ratio)
• $\hat{\phi} = \frac{1 - \sqrt{5}}{2}$ (the conjugate of the golden ratio)

Now, to compute the 23rd Fibonacci number $F_{23}$, we substitute $n = 23$ into the formula:

$F_{23} = \frac{1}{\sqrt{5}} \left( \phi^{23} - \hat{\phi}^{23} \right)$

Let's calculate this.The 23rd Fibonacci number, calculated using the Binet formula, is $F_{23} = 28,657$.

Would you like more details on how the Binet formula works, or do you have any other questions?

Here are 5 related questions to explore:

1. How is the Binet formula derived?
2. What are the limitations of using the Binet formula for very large Fibonacci numbers?
3. What is the 24th Fibonacci number?
4. Can the Fibonacci sequence be generalized to other sequences?
5. How is the Fibonacci sequence connected to the golden ratio?

Tip: The Binet formula is exact, but rounding errors can occur for very large Fibonacci numbers due to floating-point arithmetic limitations.