The 37th term of Fibonacci number using Binet’s Formula

To find the 37th term of the Fibonacci sequence using Binet's Formula, we'll use the following formula:

\[
F(n) = \frac{ \varphi^n - \psi^n }{\sqrt{5}}
\]

Where:
- \( F(n) \) is the \( n \)th Fibonacci number.
- \( \varphi = \frac{1 + \sqrt{5}}{2} \) (the golden ratio)
- \( \psi = \frac{1 - \sqrt{5}}{2} \)

Let's calculate \( F(37) \) using this formula.

\[
F(37) = \frac{ \varphi^{37} - \psi^{37} }{\sqrt{5}}
\]

Now let's compute it step by step:

1. Calculate \( \varphi^{37} \).
2. Calculate \( \psi^{37} \).
3. Subtract \( \psi^{37} \) from \( \varphi^{37} \).
4. Divide the result by \( \sqrt{5} \).

I'll calculate it for you.The 37th term of the Fibonacci sequence, calculated using Binet’s Formula, is **24,157,817**.

Would you like further details on this calculation or have any other questions? Here are five related questions you might find interesting:

1. How is Binet's Formula derived?
2. Can Binet's Formula be used for any \(n\) in the Fibonacci sequence?
3. What is the significance of the golden ratio (\(\varphi\)) in the Fibonacci sequence?
4. How does the approximation accuracy of Binet's Formula change with increasing \(n\)?
5. How can we prove that Fibonacci numbers appear in nature?

**Tip:** The Fibonacci sequence can be found in many natural patterns, such as the arrangement of leaves on a stem or the pattern of a pinecone's scales.

Learn how to calculate the 37th term of the Fibonacci sequence using Binet’s Formula. Detailed step-by-step solution involving the golden ratio and the mathematical concept of Fibonacci numbers. Suitable for advanced high school students and above.

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