Key Concepts and Formulas

• Roots of a Quadratic Equation: If $\alpha$ is a root of $ax^2 + bx + c = 0$, then $a\alpha^2 + b\alpha + c = 0$.
• Recurrence Relations: A recurrence relation defines a sequence based on previous terms. This is useful for simplifying expressions involving powers of roots.
• Factoring: Identifying common factors to simplify complex expressions.

Step-by-Step Solution

Step 1: Understand the Problem and Define Variables

We are given the quadratic equation $x^2 - x - 4 = 0$ with roots $\alpha$ and $\beta$ (where $\alpha > \beta$). We define $P_n = \alpha^n - \beta^n$ and need to find the value of $\frac{P_{15} P_{16}-P_{14} P_{16}-P_{15}^{2}+P_{14} P_{15}}{P_{13} P_{14}}$.

Why? This step clarifies the given information and sets the stage for the solution. It also defines the key term $P_n$.

Step 2: Derive the Recurrence Relation

Since $\alpha$ and $\beta$ are roots of $x^2 - x - 4 = 0$, we have: $\alpha^2 - \alpha - 4 = 0 \quad \cdots (1)$ $\beta^2 - \beta - 4 = 0 \quad \cdots (2)$

Multiply equation (1) by $\alpha^{n-1}$ and equation (2) by $\beta^{n-1}$: $\alpha^{n+1} - \alpha^n - 4\alpha^{n-1} = 0 \quad \cdots (3)$ $\beta^{n+1} - \beta^n - 4\beta^{n-1} = 0 \quad \cdots (4)$

Subtract equation (4) from equation (3): $(\alpha^{n+1} - \beta^{n+1}) - (\alpha^n - \beta^n) - 4(\alpha^{n-1} - \beta^{n-1}) = 0$ Substitute $P_n = \alpha^n - \beta^n$: $P_{n+1} - P_n - 4P_{n-1} = 0$ Therefore, the recurrence relation is: $P_{n+1} = P_n + 4P_{n-1}$ or $P_{n+1} - P_n = 4P_{n-1} \quad \cdots (5)$

Why? This step derives the crucial recurrence relation that connects different terms of the sequence $P_n$. This relation will be used to simplify the target expression.

Step 3: Apply the Recurrence Relation for Specific Terms

Using the recurrence relation $P_{n+1} - P_n = 4P_{n-1}$, we have:

For $n = 14$: $P_{15} - P_{14} = 4P_{13} \quad \cdots (6)$

For $n = 15$: $P_{16} - P_{15} = 4P_{14} \quad \cdots (7)$

Why? This step applies the derived recurrence relation to the specific terms present in the given expression, setting up the substitution in the next step.

Step 4: Simplify the Given Expression

The expression is: $\frac{P_{15} P_{16}-P_{14} P_{16}-P_{15}^{2}+P_{14} P_{15}}{P_{13} P_{14}}$

Factor the numerator: $\frac{P_{16}(P_{15} - P_{14}) - P_{15}(P_{15} - P_{14})}{P_{13} P_{14}} = \frac{(P_{15} - P_{14})(P_{16} - P_{15})}{P_{13} P_{14}}$

Substitute equations (6) and (7): $\frac{(4P_{13})(4P_{14})}{P_{13} P_{14}}$

Simplify: $\frac{16 P_{13} P_{14}}{P_{13} P_{14}} = 16$

Why? This step simplifies the complex expression using the factored form and the recurrence relations derived earlier, leading to a constant value.

Common Mistakes & Tips

• Carefully check for algebraic errors, especially when factoring.
• Remember the recurrence relation correctly; the sign is crucial.
• Ensure that you are substituting the correct values into the recurrence relation.

Summary

By deriving a recurrence relation for $P_n = \alpha^n - \beta^n$ based on the quadratic equation $x^2 - x - 4 = 0$ and substituting it into the given expression, we simplified the expression to a constant value. The key was recognizing the structure of the expression and applying the recurrence relation strategically. The final answer is 16.

Final Answer

The final answer is $\boxed{16}$.