Binet's formula provides an explicit way to calculate the $n$-th Fibonacci number. The formula is given by:

$F_n = \frac{\phi^n - \psi^n}{\sqrt{5}}$

Where:

• $\phi = \frac{1 + \sqrt{5}}{2}$ (the golden ratio)
• $\psi = \frac{1 - \sqrt{5}}{2}$
• $n$ is the Fibonacci sequence position you're calculating.

For $F_{50}$: $F_{50} = \frac{\phi^{50} - \psi^{50}}{\sqrt{5}}$

Let's calculate this step-by-step.

1. Calculate $\phi$ and $\psi$: $\phi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887$ $\psi = \frac{1 - \sqrt{5}}{2} \approx -0.6180339887$

2. Raise $\phi$ and $\psi$ to the 50th power: $\phi^{50} \approx (1.6180339887)^{50}$ $\psi^{50} \approx (-0.6180339887)^{50}$

3. Subtract $\psi^{50}$ from $\phi^{50}$ and divide by $\sqrt{5}$.

Calculating this directly using Binet's formula:

$F_{50} = \frac{(1.6180339887)^{50} - (-0.6180339887)^{50}}{\sqrt{5}}$

Let me compute the value.Using Binet's formula, $F_{50}$ is calculated to be:

$F_{50} = 12,586,269,025$

This is the 50th Fibonacci number.

Would you like a more detailed breakdown or have any questions?

Here are some related questions you might find interesting:

1. How is Binet's formula derived from the Fibonacci sequence?
2. What are some applications of Fibonacci numbers in real life?
3. How does the golden ratio ($\phi$) relate to Fibonacci numbers?
4. What is the significance of $\psi$ in Binet's formula?
5. How can we approximate large Fibonacci numbers without using Binet's formula?

Tip: Binet's formula is an elegant way to calculate Fibonacci numbers directly without needing to generate all the preceding numbers, making it especially useful for large indices.