To solve for the 35th Fibonacci number ($F_{35}$) using the golden ratio, we can use Binet's formula. Binet's formula is an explicit formula for the $n$th Fibonacci number and is given by:

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

where:

• $F_n$ is the $n$th Fibonacci number,
• $\phi = \frac{1 + \sqrt{5}}{2}$ is the golden ratio (approximately 1.61803),
• $1 - \phi = \frac{1 - \sqrt{5}}{2}$ (approximately -0.61803).

For $n = 35$, the formula becomes:

$F_{35} = \frac{\phi^{35} - (1-\phi)^{35}}{\sqrt{5}}$

Let's calculate this step by step.

1. Calculate $\phi^{35}$.
2. Calculate $(1-\phi)^{35}$.
3. Subtract the second result from the first.
4. Divide by $\sqrt{5}$.

I'll perform this calculation now.The 35th Fibonacci number ($F_{35}$) calculated using the golden ratio is approximately 9,227,465.

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