Binet's formula for Fibonacci numbers is given by:

\[
F_n = \frac{{\phi^n - (1-\phi)^n}}{\sqrt{5}}
\]

where \( \phi = \frac{{1 + \sqrt{5}}}{2} \) (the golden ratio).

We need to find \( 2F_{25} + F_{30} \). Using Binet’s formula, the individual Fibonacci numbers are:

\[
F_{25} = \frac{{\phi^{25} - (1 - \phi)^{25}}}{\sqrt{5}}
\]
\[
F_{30} = \frac{{\phi^{30} - (1 - \phi)^{30}}}{\sqrt{5}}
\]

Thus, the expression \( 2F_{25} + F_{30} \) becomes:

\[
2F_{25} + F_{30} = 2\left(\frac{{\phi^{25} - (1-\phi)^{25}}}{\sqrt{5}}\right) + \frac{{\phi^{30} - (1-\phi)^{30}}}{\sqrt{5}}
\]

\[
= \frac{{2\phi^{25} - 2(1-\phi)^{25}}}{\sqrt{5}} + \frac{{\phi^{30} - (1-\phi)^{30}}}{\sqrt{5}}
\]

\[
= \frac{{2\phi^{25} + \phi^{30} - 2(1-\phi)^{25} - (1-\phi)^{30}}}{\sqrt{5}}
\]

Now, we can simplify this formula numerically using approximations of \( \phi \) and \( 1-\phi \). I'll calculate this next.The value of \( 2F_{25} + F_{30} \) is approximately 982,090.

Would you like more details on the calculation, or do you have any questions?

Here are some related questions to consider:
1. How does Binet's formula relate to the recursive definition of Fibonacci numbers?
2. What are the limitations of using Binet’s formula for large Fibonacci numbers?
3. How is the golden ratio (\(\phi\)) involved in Fibonacci sequences and other areas of mathematics?
4. Can Binet’s formula be applied to non-integer Fibonacci numbers?
5. What are some alternative methods to calculate large Fibonacci numbers?

**Tip:** When calculating Fibonacci numbers for large indices, numerical methods like Binet's formula can be prone to rounding errors due to the limitations of floating-point arithmetic.

Solve the problem 2F25 + F30 using Binet's Formula and the golden ratio. This detailed explanation covers Fibonacci numbers, Binet's formula, and its application for large Fibonacci terms, suitable for advanced high school and college-level students.

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