The Binet formula provides a way to compute any Fibonacci number without needing to calculate all preceding terms. The formula is expressed as:

$F_n = \frac{1}{\sqrt{5}} \left( \left( \frac{1 + \sqrt{5}}{2} \right)^n - \left( \frac{1 - \sqrt{5}}{2} \right)^n \right)$

To find $F_{50}$, plug in $n = 50$:

$F_{50} = \frac{1}{\sqrt{5}} \left( \left( \frac{1 + \sqrt{5}}{2} \right)^{50} - \left( \frac{1 - \sqrt{5}}{2} \right)^{50} \right)$

Since $\left( \frac{1 - \sqrt{5}}{2} \right)^{50}$ is very small (close to zero), the formula simplifies to:

$F_{50} \approx \frac{1}{\sqrt{5}} \left( \left( \frac{1 + \sqrt{5}}{2} \right)^{50} \right)$

Let's calculate this using approximate values for the constants:

• $\frac{1 + \sqrt{5}}{2} \approx 1.6180339887$ (the golden ratio $\phi$)
• $\frac{1 - \sqrt{5}}{2} \approx -0.6180339887$

Thus, $F_{50} \approx \frac{1.6180339887^{50}}{\sqrt{5}}$.

Now, I will compute $F_{50}$.

The 50th Fibonacci number $F_{50}$ using Binet's formula is approximately:

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

Would you like more details on the calculation, or do you have any questions?

Here are five related questions:

1. What is the Fibonacci sequence, and how is it constructed?
2. Can you derive the Binet formula step by step?
3. How does the Binet formula approximate large Fibonacci numbers?
4. What are some applications of the Fibonacci sequence in real life?
5. How is the golden ratio $\phi$ related to Fibonacci numbers?

Tip: The Fibonacci sequence grows exponentially, and the golden ratio $\phi$ plays a key role in understanding this growth!