To find the 50th and 75th terms of the Fibonacci sequence using Binet's formula, we'll proceed step by step.

Binet's Formula

The formula for the $n$-th term $F(n)$ of the Fibonacci sequence is given by:

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

where:

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

Finding the 50th Term $F(50)$

Using Binet's formula:

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

Since $\psi$ is a negative number less than 1, $\psi^{50}$ will be very close to 0, so it can be approximated as:

$F(50) \approx \frac{\phi^{50}}{\sqrt{5}}$

Let's calculate $F(50)$:

1. Calculate $\phi^{50}$: $\phi^{50} \approx 1.6180339887^{50}$

2. Approximate $\psi^{50}$: \psi^{50} \approx (-0.6180339887)^{50} \approx 0 \text{ (since \psi^{50} is extremely small)}

3. Calculate $F(50)$: $F(50) \approx \frac{\phi^{50}}{\sqrt{5}}$

Let's approximate this using the values:

$\phi^{50} \approx 28143753123.000002$ $\sqrt{5} \approx 2.2360679775$ $F(50) \approx \frac{28143753123.000002}{2.2360679775} \approx 12586269025$

Finding the 75th Term $F(75)$

Similarly, for the 75th term:

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

Again, $\psi^{75}$ will be very close to 0, so:

$F(75) \approx \frac{\phi^{75}}{\sqrt{5}}$

Let's calculate $F(75)$:

1. Calculate $\phi^{75}$: $\phi^{75} \approx 1.6180339887^{75}$

2. Approximate $\psi^{75}$: $\psi^{75} \approx (-0.6180339887)^{75} \approx 0$

3. Calculate $F(75)$: $F(75) \approx \frac{\phi^{75}}{\sqrt{5}}$

Let's approximate this using the values:

$\phi^{75} \approx 17167680177565.025$ $\sqrt{5} \approx 2.2360679775$ $F(75) \approx \frac{17167680177565.025}{2.2360679775} \approx 767761997860$

Final Answers:

• The 50th term of the Fibonacci sequence is approximately 12,586,269,025.
• The 75th term of the Fibonacci sequence is approximately 767,761,997,860.

Would you like more details or have any other questions?

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Here are some related questions:

1. How does Binet’s formula relate to the Fibonacci sequence?
2. What is the significance of the golden ratio $\phi$ in Fibonacci numbers?
3. Can Binet’s formula be used for non-integer values of $n$?
4. What is the limit of the ratio $F(n+1)/F(n)$ as $n$ approaches infinity?
5. How would the solution change if the Fibonacci sequence started with different initial values?

Tip: In calculations involving large Fibonacci numbers, it's useful to recognize that $\psi^n$ becomes negligible as $n$ increases, simplifying the computation.