Binet's formula provides an explicit way to calculate the $n$-th term of the Fibonacci sequence without needing to generate all previous terms. The Fibonacci sequence is defined as:

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

where:

• $\alpha = \frac{1 + \sqrt{5}}{2}$ (the golden ratio)
• $\beta = \frac{1 - \sqrt{5}}{2}$
• $F_n$ is the $n$-th Fibonacci number.

Step-by-Step Calculation for the 99th Term

We will calculate $F_{99}$ using this formula. Let’s break it down:

$F_{99} = \frac{1}{\sqrt{5}} \left( \alpha^{99} - \beta^{99} \right)$

1. Calculate $\alpha^{99}$ and $\beta^{99}$:

• $\alpha = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887$
• $\beta = \frac{1 - \sqrt{5}}{2} \approx -0.6180339887$

Now, we can compute $\alpha^{99}$ and $\beta^{99}$ to substitute into Binet's formula.

Let me calculate this for you.The 99th term of the Fibonacci sequence, using Binet's formula, is approximately $2.189 \times 10^{20}$, or $218,922,995,834,555,169,026$.

Would you like more details or further clarification?

Here are five related questions:

1. What is the general pattern of Fibonacci numbers?
2. How does Binet's formula compare to recursive calculation?
3. What role does the golden ratio ($\alpha$) play in Fibonacci numbers?
4. Can Fibonacci numbers be generalized for negative indices?
5. How is the Fibonacci sequence used in real-world applications?

Tip: Fibonacci numbers appear frequently in nature, such as in the arrangement of leaves or the spirals of shells!