The common formula to calculate the \(n\)th Fibonacci number is known as Binet's Formula, which is:

\[
F(n) = \frac{\phi^n - (1-\phi)^n}{\sqrt{5}}
\]

where \(\phi\) (the golden ratio) is given by:

\[
\phi = \frac{1 + \sqrt{5}}{2}
\]

Now, let's apply Binet's formula to find \( F(23) \):

\[
F(23) = \frac{\left(\frac{1 + \sqrt{5}}{2}\right)^{23} - \left(\frac{1 - \sqrt{5}}{2}\right)^{23}}{\sqrt{5}}
\]

This simplifies to:

\[
F(23) = \frac{\phi^{23} - (1-\phi)^{23}}{\sqrt{5}}
\]

Given that \(\phi\) is approximately 1.618 and \(1-\phi\) is approximately -0.618, the second term becomes very small, and the calculation becomes:

\[
F(23) \approx \frac{(1.618)^{23} - (-0.618)^{23}}{\sqrt{5}}
\]

Calculating this gives:

\[
F(23) \approx \frac{28,657.12 - 0.12}{2.236} \approx \frac{28,657}{2.236} \approx 28,657
\]

So, \( F(23) = 28,657 \), confirming our earlier calculation.

Would you like to explore this further or have any questions?

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Here are five related questions to expand on this topic:

1. How can Binet's formula be derived from the Fibonacci recurrence relation?
2. How accurate is Binet's formula for very large \( n \)?
3. What is the significance of the golden ratio in the derivation of Binet's formula?
4. How does the Fibonacci sequence relate to continued fractions?
5. How is the Fibonacci sequence applied in algorithmic complexity analysis?

**Tip:** Binet's formula is exact for Fibonacci numbers but can be challenging to compute manually for large \(n\) due to the powers involved.

Explore how to calculate the 23rd Fibonacci number using Binet's Formula and the significance of the golden ratio. Binet's Formula provides an exact method for computing Fibonacci numbers, leveraging the mathematical properties of the golden ratio. Learn more about its derivation and accuracy.

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