Here are the  the additional activity, from the given Fibonacci sequence  0,1,1,2,3,5,8,11,13.......Find the 50th term and 75th term using Binets formula , show your solution 

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?

---

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.

Here are the additional activity, from the given Fibonacci sequence  0,1,1,2,3,5,8,11,13.......Find the 50th term and 75th term using Binet's formula, show your solution

Discover how to find the 50th and 75th terms of the Fibonacci sequence using Binet's formula. This detailed solution covers key mathematical concepts such as the Fibonacci sequence, the golden ratio, and exponential functions. Ideal for students in Grades 10-12.

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