To solve \( 2F_{25} + F_{30} \) using Binet's formula, we first need to understand the general form of the Fibonacci sequence, denoted \( F_n \), in terms of Binet's formula:

\[
F_n = \frac{1}{\sqrt{5}} \left( \alpha^n - \beta^n \right)
\]
where:
\[
\alpha = \frac{1 + \sqrt{5}}{2}, \quad \beta = \frac{1 - \sqrt{5}}{2}
\]

Now, we can calculate \( F_{25} \) and \( F_{30} \), and then solve \( 2F_{25} + F_{30} \).

### Step 1: Calculate \( F_{25} \)
\[
F_{25} = \frac{1}{\sqrt{5}} \left( \alpha^{25} - \beta^{25} \right)
\]

### Step 2: Calculate \( F_{30} \)
\[
F_{30} = \frac{1}{\sqrt{5}} \left( \alpha^{30} - \beta^{30} \right)
\]

### Step 3: Combine results for \( 2F_{25} + F_{30} \)
The expression becomes:
\[
2F_{25} + F_{30} = 2 \times \frac{1}{\sqrt{5}} \left( \alpha^{25} - \beta^{25} \right) + \frac{1}{\sqrt{5}} \left( \alpha^{30} - \beta^{30} \right)
\]
\[
= \frac{1}{\sqrt{5}} \left( 2\alpha^{25} - 2\beta^{25} + \alpha^{30} - \beta^{30} \right)
\]

Next, we compute the values of \( \alpha^{25}, \beta^{25}, \alpha^{30}, \beta^{30} \), and substitute them back into the equation to obtain the result.

Let me calculate this for you.The value of \( 2F_{25} + F_{30} \) is approximately 982,090.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How does Binet's formula approximate Fibonacci numbers for large \( n \)?
2. Why does the term \( \beta^n \) become insignificant as \( n \) increases?
3. Can Binet’s formula be used to calculate Fibonacci numbers in a computationally efficient way?
4. What is the relation between Fibonacci numbers and the golden ratio \( \alpha \)?
5. How does the Fibonacci sequence appear in nature and other mathematical contexts?

**Tip**: Binet's formula gives very accurate results for large Fibonacci numbers, but rounding errors can still occur due to floating-point precision limits.

Learn how to solve the Fibonacci expression 2F25 + F30 using Binet's formula, which involves the golden ratio. This problem includes detailed calculations using Binet's theorem to derive large Fibonacci numbers and is suitable for advanced high school or college students.

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