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.