Find the 99th term of Fibonacci Sequence Using Binet's Formula

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!

Learn how to calculate the 99th term of the Fibonacci sequence using Binet's formula. This solution involves key mathematical concepts such as the Fibonacci sequence, the golden ratio, and explicit formula calculation. Suitable for high school students in Grades 9-12.

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