what is the 35th term in Fiboncci secquence using Binet's formula

Binet's formula provides an explicit way to calculate the \(n\)th term of the Fibonacci sequence without having to compute all the preceding terms. The formula is given by:

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

where:

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

To find the 35th term in the Fibonacci sequence (\(F(35)\)), we can plug \(n = 35\) into the formula:

\[
F(35) = \frac{\left(\frac{1 + \sqrt{5}}{2}\right)^{35} - \left(\frac{1 - \sqrt{5}}{2}\right)^{35}}{\sqrt{5}}
\]

Let's calculate this.

First, approximate the values:

- \(\phi \approx 1.6180339887\)
- \(1 - \phi \approx -0.6180339887\)
- \(\sqrt{5} \approx 2.236067977\)

Then calculate:

\[
F(35) \approx \frac{1.6180339887^{35} - (-0.6180339887)^{35}}{2.236067977}
\]

Let's compute the value.The 35th term in the Fibonacci sequence, calculated using Binet's formula, is approximately \( F(35) = 9,227,465 \).

Would you like more details on the calculation, or do you have any other questions? Here are some related questions to consider:

1. What is Binet's formula used for, and how does it work?
2. How does the golden ratio relate to the Fibonacci sequence?
3. Can Binet's formula be used for any Fibonacci number, and are there any limitations?
4. How does the precision of Binet's formula compare to iterative methods?
5. What are some applications of Fibonacci numbers in real-world scenarios?

**Tip:** When using Binet's formula for larger Fibonacci numbers, the formula's approximation can be slightly off due to rounding errors, but it is still quite accurate for practical purposes.

Discover how to compute the 35th term in the Fibonacci sequence using Binet's Formula. This formula leverages the properties of the golden ratio and exponential growth to provide an accurate result. Explore the step-by-step calculation and understand the application of Binet's Formula in Fibonacci sequences.

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