Evaluate Twelfth Term in Sequence Defined by Quadratic Roots

Given that $\alpha$ and $\beta$ are the distinct roots of the quadratic equation: $x^2 - x - 1 = 0$

Since $\alpha$ and $\beta$ satisfy this equation, we have: $\alpha^2 - \alpha - 1 = 0 \implies \alpha^2 = \alpha + 1$ $\beta^2 - \beta - 1 = 0 \implies \beta^2 = \beta + 1$

Multiplying both sides of the equations by $\alpha^n$ and $\beta^n$ respectively for any non-negative integer $n$, we get: $\alpha^{n+2} = \alpha^{n+1} + \alpha^n$ $\beta^{n+2} = \beta^{n+1} + \beta^n$

Now, multiplying the first equation by $p$ and the second equation by $q$, and adding them together yields: $p \alpha^{n+2} + q \beta^{n+2} = (p \alpha^{n+1} + q \beta^{n+1}) + (p \alpha^n + q \beta^n)$

Using the definition $a_n = p \alpha^n + q \beta^n$, this recurrence relation simplifies to: $a_{n+2} = a_{n+1} + a_n$

To find $a_{12}$, we substitute $n = 10$ into the recurrence relation: $a_{12} = a_{11} + a_{10}$

Thus, the correct option is B.