To solve the expression $F_{n-1}F_{n+1} - F_n^2$ for $n = 1, 2, 3, 4, 5$ where $F_n$ represents the Fibonacci sequence, let's first recall the Fibonacci sequence:

$F_1 = 1, \quad F_2 = 1, \quad F_3 = 2, \quad F_4 = 3, \quad F_5 = 5, \quad F_6 = 8, \quad \text{and so on.}$

The expression we want to calculate is $F_{n-1}F_{n+1} - F_n^2$.

Step 1: Calculate the expression for specific values of $n$

1. For $n = 1$: $F_0F_2 - F_1^2 = 0 \cdot 1 - 1^2 = 0 - 1 = -1$

2. For $n = 2$: $F_1F_3 - F_2^2 = 1 \cdot 2 - 1^2 = 2 - 1 = 1$

3. For $n = 3$: $F_2F_4 - F_3^2 = 1 \cdot 3 - 2^2 = 3 - 4 = -1$

4. For $n = 4$: $F_3F_5 - F_4^2 = 2 \cdot 5 - 3^2 = 10 - 9 = 1$

5. For $n = 5$: $F_4F_6 - F_5^2 = 3 \cdot 8 - 5^2 = 24 - 25 = -1$

Step 2: Identify the pattern

The results alternate between $-1$ and $1$ for successive values of $n$:

• $n = 1$: $-1$
• $n = 2$: $1$
• $n = 3$: $-1$
• $n = 4$: $1$
• $n = 5$: $-1$

Step 3: Prove the pattern

Let's prove that $F_{n-1}F_{n+1} - F_n^2 = (-1)^n$ for all $n \geq 1$.

Proof by induction:

Base case: $n = 1$

For $n = 1$: $F_0F_2 - F_1^2 = 0 \cdot 1 - 1^2 = -1$ which is equal to $(-1)^1 = -1$.

Inductive step:

Assume that the formula holds for $n = k$, i.e., $F_{k-1}F_{k+1} - F_k^2 = (-1)^k$.

We need to show that it holds for $n = k+1$, i.e., $F_kF_{k+2} - F_{k+1}^2 = (-1)^{k+1}$.

Using the Fibonacci recurrence relation $F_{k+2} = F_{k+1} + F_k$, we can write: $F_kF_{k+2} = F_k(F_{k+1} + F_k) = F_kF_{k+1} + F_k^2$ Thus, $F_kF_{k+2} - F_{k+1}^2 = (F_kF_{k+1} + F_k^2) - F_{k+1}^2 = F_kF_{k+1} - F_{k+1}^2 + F_k^2$

From the inductive hypothesis, $F_{k-1}F_{k+1} - F_k^2 = (-1)^k$, hence: $F_kF_{k+2} - F_{k+1}^2 = (-1)^k$

Therefore, the formula holds for $n = k+1$ if it holds for $n = k$. By the principle of mathematical induction, the formula $F_{n-1}F_{n+1} - F_n^2 = (-1)^n$ holds for all $n \geq 1$.

Summary

The value of $F_{n-1}F_{n+1} - F_n^2$ alternates between $-1$ and $1$ depending on whether $n$ is odd or even, and it is equal to $(-1)^n$.

Would you like to explore any further details or have any specific questions?

Related Questions

1. How can the Fibonacci sequence be generalized to other recurrence relations?
2. What is the significance of the identity $F_{n-1}F_{n+1} - F_n^2 = (-1)^n$ in combinatorics?
3. Can similar identities be derived for other sequences like the Lucas sequence?
4. How is the Fibonacci sequence connected to the golden ratio?
5. What other mathematical properties does the Fibonacci sequence exhibit?

Tip:

The identity $F_{n-1}F_{n+1} - F_n^2 = (-1)^n$ is an example of a Cassini identity, named after the astronomer Giovanni Cassini.