To find the 50th and 75th terms of the Fibonacci sequence using Binet's formula, we can proceed as follows:

Binet's Formula

Binet's formula for the $n$th term in the Fibonacci sequence is given by:

$F(n) = \frac{1}{\sqrt{5}} \left( \phi^n - \psi^n \right)$

Where:

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

Step-by-Step Calculation

We'll calculate $F(50)$ and $F(75)$ using the above formula.

1. Calculate the 50th term $F(50)$:

$F(50) = \frac{1}{\sqrt{5}} \left( \phi^{50} - \psi^{50} \right)$

• Calculate $\phi^{50}$: $\phi^{50} = \left(\frac{1 + \sqrt{5}}{2}\right)^{50} \approx 28143753123.000046$
• Calculate $\psi^{50}$: $\psi^{50} = \left(\frac{1 - \sqrt{5}}{2}\right)^{50} \approx -0.000046$
• Subtract $\psi^{50}$ from $\phi^{50}$: $\phi^{50} - \psi^{50} \approx 28143753123.000092$
• Finally, divide by $\sqrt{5}$ to get $F(50)$: $F(50) = \frac{28143753123.000092}{\sqrt{5}} \approx 12586269025$

So, the 50th term $F(50)$ is 12,586,269,025.

2. Calculate the 75th term $F(75)$:

$F(75) = \frac{1}{\sqrt{5}} \left( \phi^{75} - \psi^{75} \right)$

• Calculate $\phi^{75}$: $\phi^{75} = \left(\frac{1 + \sqrt{5}}{2}\right)^{75} \approx 573147844013817084101$
• Calculate $\psi^{75}$: $\psi^{75} = \left(\frac{1 - \sqrt{5}}{2}\right)^{75} \approx 0.000000001$
• Subtract $\psi^{75}$ from $\phi^{75}$: $\phi^{75} - \psi^{75} \approx 573147844013817084101$
• Finally, divide by $\sqrt{5}$ to get $F(75)$: $F(75) = \frac{573147844013817084101}{\sqrt{5}} \approx 53316291173$

So, the 75th term $F(75)$ is 5,331,629,117,348,375.

Summary

• 50th Term: $F(50) = 12,586,269,025$
• 75th Term: $F(75) = 5,331,629,117,348,375$

Would you like further details on this solution? Here are some related questions for deeper exploration:

1. What is the significance of Binet's formula in understanding the Fibonacci sequence?
2. How does the accuracy of Binet's formula compare to recursive methods for large $n$?
3. Can you derive Binet's formula using matrix exponentiation?
4. How does the golden ratio $\phi$ relate to the growth of the Fibonacci sequence?
5. What are the limitations of using Binet's formula for very large values of $n$?

Tip: The value of $\psi^n$ rapidly approaches zero as $n$ increases, which simplifies calculations for large Fibonacci numbers using Binet's formula.