Calculate The 64th term of Fibonacci number using Binet’s Formula

To calculate the 64th term of the Fibonacci sequence using Binet’s Formula, we can proceed with the following steps. Binet's formula for the nth Fibonacci number is given by:

\[
F_n = \frac{\phi^n - \psi^n}{\sqrt{5}}
\]

where:

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

### Step 1: Assign the values
\[
\phi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887
\]
\[
\psi = \frac{1 - \sqrt{5}}{2} \approx -0.6180339887
\]

### Step 2: Apply the formula to calculate \( F_{64} \)

Substituting \( n = 64 \) into Binet’s formula:

\[
F_{64} = \frac{\phi^{64} - \psi^{64}}{\sqrt{5}}
\]

### Step 3: Calculate \(\phi^{64}\) and \(\psi^{64}\)

Using the approximations of \(\phi\) and \(\psi\):

\[
\phi^{64} \approx (1.6180339887)^{64}
\]
\[
\psi^{64} \approx (-0.6180339887)^{64}
\]

Calculating these:

\[
\phi^{64} \approx 6.2351915 \times 10^{12}
\]
\[
\psi^{64} \approx 1.6039864 \times 10^{-13}
\]

### Step 4: Substitute the values back into the formula

\[
F_{64} = \frac{6.2351915 \times 10^{12} - 1.6039864 \times 10^{-13}}{\sqrt{5}}
\]

Since \( \psi^{64} \) is very close to zero, it can be approximated to zero for simplicity in large Fibonacci number calculations:

\[
F_{64} \approx \frac{6.2351915 \times 10^{12}}{\sqrt{5}}
\]

\[
\approx \frac{6.2351915 \times 10^{12}}{2.236067977}
\]

\[
\approx 2,385,973,078,601
\]

### Conclusion

The 64th term of the Fibonacci sequence, calculated using Binet's formula, is approximately:

\[
F_{64} \approx 2,385,973,078,601
\]

### Would you like more details or have any questions?

Here are some related questions to explore:

1. What is the 65th Fibonacci number using Binet's formula?
2. How does Binet's formula compare in accuracy with the recursive method for large Fibonacci numbers?
3. How is the golden ratio related to Fibonacci numbers?
4. Can Binet’s formula be used for negative values of \(n\)? If so, how?
5. How can we derive Binet’s formula from the Fibonacci sequence?

**Tip:** For large Fibonacci numbers, using approximations such as Binet's formula can be useful for quick estimations, but for exact values, more precise computational techniques may be necessary.

Learn how to calculate the 64th term of the Fibonacci sequence using Binet’s Formula. Includes step-by-step instructions and computations using the golden ratio and its conjugate. Explore the accuracy of Binet’s Formula for large Fibonacci numbers. Suitable for advanced high school students.

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