To find the value of the 35th Fibonacci number ($F_{35}$) using Binet's formula, we use the following expression:

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

where:

• $\phi = \frac{1 + \sqrt{5}}{2}$ is the golden ratio,
• $(1 - \phi) = \frac{1 - \sqrt{5}}{2}$.

Substituting $n = 35$ into the formula, we get:

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

Let’s calculate this step by step:

1. Calculate $\phi$ and $1 - \phi$.
2. Raise both $\phi$ and $1 - \phi$ to the power of 35.
3. Subtract the result of $(1 - \phi)^{35}$ from $\phi^{35}$.
4. Divide the result by $\sqrt{5}$.

Let's compute this value.The value of $F_{35}$ using Binet's formula is approximately $9,227,465$.

Would you like more details or have any questions? Here are some related questions that might interest you:

1. How does Binet's formula work for calculating Fibonacci numbers?
2. What is the golden ratio, and why is it important in Fibonacci numbers?
3. How can we approximate Fibonacci numbers without using Binet's formula?
4. What are some applications of Fibonacci numbers in real life?
5. How does the value of $F_{35}$ compare to the values of other Fibonacci numbers?

Tip: Binet's formula is exact for Fibonacci numbers when using infinite precision, but when implemented on a computer, minor inaccuracies may arise due to floating-point arithmetic, especially for large indices.