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The image shows a problem asking for the 25th term of the Fibonacci sequence, using the closed-form expression known as **Binet's formula**:

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

Where:
- \( F_n \) is the nth Fibonacci number,
- \( \frac{1 + \sqrt{5}}{2} \) is known as the golden ratio \( \phi \),
- \( \frac{1 - \sqrt{5}}{2} \) is often denoted as \( \psi \), and
- \( \sqrt{5} \) is the square root of 5.

### Step-by-Step Solution:

For \( n = 25 \), we substitute the value into Binet's formula.

\[
F_{25} = \frac{1}{\sqrt{5}} \left( \left( \frac{1 + \sqrt{5}}{2} \right)^{25} - \left( \frac{1 - \sqrt{5}}{2} \right)^{25} \right)
\]

I will now compute the value of \( F_{25} \).The 25th term of the Fibonacci sequence is \( F_{25} = 75,025 \).

Would you like further details on how this was calculated or have any other questions?

Here are 5 related questions to explore:

1. What is Binet’s formula used for in general?
2. How does the Fibonacci sequence grow as \( n \) increases?
3. Can Fibonacci numbers be calculated using other methods besides Binet's formula?
4. What are the real-world applications of Fibonacci numbers?
5. How is the golden ratio related to Fibonacci numbers?

**Tip**: Binet's formula provides an exact solution, but for large \( n \), approximation techniques might be more practical due to computational limitations.

Find the 25th term of the Fibonacci numbers?

Calculate the 25th term of the Fibonacci sequence using Binet's formula. This problem covers key number theory concepts and involves calculating the golden ratio. Suitable for undergraduate students interested in number theory and sequences.

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