The expression you've provided represents the 50th Fibonacci number $F_{50}$, using Binet's formula. Let's break it down:

Binet's Formula for the Fibonacci Sequence:

$F_n = \frac{\phi^n - (1 - \phi)^n}{\sqrt{5}}$ where:

• $\phi$ is the golden ratio, given by $\phi = \frac{1 + \sqrt{5}}{2}$,
• $n$ is the position of the Fibonacci number in the sequence (in your case, $n = 50$).

Now, for $F_{50}$: $F_{50} = \frac{\phi^{50} - (1 - \phi)^{50}}{\sqrt{5}}$ This formula works due to the rapid decay of $(1 - \phi)^n$, making it negligible for large $n$, so the dominant term is $\phi^n$.

Would you like me to calculate the exact value of $F_{50}$ using this formula?

Here are five related questions to expand the topic:

1. What is the significance of the golden ratio $\phi$ in the Fibonacci sequence?
2. Can you derive the approximation of $F_n$ for large $n$ based on Binet's formula?
3. How does the second term $(1 - \phi)^n$ contribute to the accuracy of the formula?
4. How does the growth rate of Fibonacci numbers relate to the golden ratio?
5. What are some real-world applications of the Fibonacci sequence?

Tip: The Fibonacci sequence appears in many natural phenomena, such as the arrangement of leaves on a stem or the spirals of shells.