Calculate The 100th term of Fibonacci number using Binet’s Formula

To calculate the 100th term of the Fibonacci sequence using Binet’s formula, we proceed similarly to the method used for the 64th term. Binet’s formula is given by:

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

where:

- \(\phi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887\) (the golden ratio),
- \(\psi = \frac{1 - \sqrt{5}}{2} \approx -0.6180339887\).

### Step 1: Set \( n = 100 \) in Binet’s Formula

\[
F_{100} = \frac{\phi^{100} - \psi^{100}}{\sqrt{5}}
\]

### Step 2: Calculate \(\phi^{100}\) and \(\psi^{100}\)

Using approximations for \(\phi\) and \(\psi\):

\[
\phi^{100} \approx (1.6180339887)^{100}
\]
\[
\psi^{100} \approx (-0.6180339887)^{100}
\]

Calculating these:

- \(\phi^{100}\) will be a very large number, whereas \(\psi^{100}\) will be very small since \(\psi\) is less than 1 in absolute value.

\[
\phi^{100} \approx 7.9207084 \times 10^{20}
\]
\[
\psi^{100} \approx 2.5151513 \times 10^{-21}
\]

### Step 3: Substitute Values into Binet’s Formula

\[
F_{100} = \frac{7.9207084 \times 10^{20} - 2.5151513 \times 10^{-21}}{\sqrt{5}}
\]

Since \(\psi^{100}\) is very close to zero, we can approximate it as zero for practical purposes:

\[
F_{100} \approx \frac{7.9207084 \times 10^{20}}{\sqrt{5}}
\]

\[
\approx \frac{7.9207084 \times 10^{20}}{2.236067977}
\]

\[
\approx 3.542248481792631 \times 10^{20}
\]

### Step 4: Calculate the Approximation

\[
F_{100} \approx 354,224,848,179,263,111,168
\]

### Conclusion

The 100th term of the Fibonacci sequence, calculated using Binet's formula, is approximately:

\[
F_{100} \approx 354,224,848,179,263,111,168
\]

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

Here are some related questions to explore:

1. What is the exact 100th Fibonacci number without approximation using a computer program?
2. How accurate is Binet's formula for calculating Fibonacci numbers for very large \( n \)?
3. How does Binet’s formula relate to the closed-form expression for the Fibonacci sequence?
4. What are the numerical methods to compute large Fibonacci numbers efficiently?
5. Can Binet’s formula be derived using matrix exponentiation methods?

**Tip:** Binet's formula is a great theoretical tool to understand the growth rate of the Fibonacci sequence, but for exact computations, especially with very large \( n \), it is often more practical to use iterative or matrix methods due to precision limitations in calculations involving irrational numbers.

Learn how to calculate the 100th term of the Fibonacci sequence using Binet’s formula. Binet’s formula involves the use of the golden ratio and provides an approximate solution for large Fibonacci numbers. Explore step-by-step calculations and the accuracy of Binet's formula.

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