Determine whether each of the following statements about Fibonacci numbers is true or false.

a. F1+F2+...Fn=Fn+2-1 -

b. 2Fn-Fn-1= Fn+1 for n >= 3 -

c. 2Fn>Fn+1 for n>=3 -

d. F3n is an even number -

e. Fn and Fn+1 for n>= 3 are relatively prime.*

Let's analyze each statement involving Fibonacci numbers to determine whether it's true or false.

### a. \( F_1 + F_2 + \dots + F_n = F_{n+2} - 1 \)

This statement is **true**. It's a well-known identity in Fibonacci sequences. The sum of the first \( n \) Fibonacci numbers is equal to the \((n+2)\)th Fibonacci number minus 1. This can be proved by induction.

### b. \( 2F_n - F_{n-1} = F_{n+1} \) for \( n \geq 3 \)

Let's verify this using the Fibonacci sequence definition \( F_{n+1} = F_n + F_{n-1} \).

\[
2F_n - F_{n-1} = F_{n+1} 
\]

Substitute the right side:
\[
F_{n+1} = F_n + F_{n-1}
\]

So the equation becomes:
\[
2F_n - F_{n-1} = F_n + F_{n-1}
\]

Simplifying:
\[
F_n = 2F_{n-1}
\]

This statement is clearly **false** as it is not a correct identity for Fibonacci numbers.

### c. \( 2F_n > F_{n+1} \) for \( n \geq 3 \)

Let's check this using specific Fibonacci numbers:

- For \( n = 3 \), \( F_3 = 2 \), \( F_4 = 3 \), and \( 2F_3 = 4 \). So, \( 4 > 3 \), which is true.
- For \( n = 4 \), \( F_4 = 3 \), \( F_5 = 5 \), and \( 2F_4 = 6 \). So, \( 6 > 5 \), which is true.

This holds for \( n \geq 3 \), so this statement is **true**.

### d. \( F_{3n} \) is an even number

The Fibonacci sequence alternates between even and odd numbers starting from \( F_3 \). Every third Fibonacci number is even:

- \( F_3 = 2 \), \( F_6 = 8 \), \( F_9 = 34 \), etc.

Thus, this statement is **true**.

### e. \( F_n \) and \( F_{n+1} \) for \( n \geq 3 \) are relatively prime

This is a known property of Fibonacci numbers. Any two consecutive Fibonacci numbers are relatively prime, meaning their greatest common divisor (gcd) is 1. Therefore, this statement is **true**.

### Summary:

- **a. True**
- **b. False**
- **c. True**
- **d. True**
- **e. True**

Would you like more details on any of these results, or do you have any questions?

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Here are 5 related questions you might find interesting:

1. How can the Fibonacci sequence be represented using matrix exponentiation?
2. Can you prove the identity \( F_1 + F_2 + \dots + F_n = F_{n+2} - 1 \) using mathematical induction?
3. What are some applications of the Fibonacci sequence in nature and art?
4. How does the ratio of consecutive Fibonacci numbers relate to the golden ratio?
5. Can you derive a formula for the nth Fibonacci number using the closed-form expression known as Binet's formula?

**Tip:** Fibonacci numbers have many interesting properties and identities, including connections to the golden ratio and applications in various fields such as computer science, biology, and art.

Explore the truth behind statements involving Fibonacci numbers, including identities and properties. Analyze statements such as sum identities, inequalities, and divisibility rules. Suitable for advanced high school students and math enthusiasts.

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