Question 127
It is known that $p, q$ ($q \neq 0$) are real numbers; the equation $x^2 - px + q = 0$ has two real roots $\alpha, \beta$; the sequence $\{a_n\}$ satisfies $a_1 = p$, $a_2 = p^2 - q$, $a_n = p a_{n-1} - q a_{n-2}$ ($n = 3, 4, \dots$).
a. Find the general expression of $\{a_n\}$ in terms of $\alpha, \beta$.
b. If $p = 1$, $q = \frac{1}{4}$, find the sum of the first $n$ terms of $\{a_n\}$.
a. Find the general expression of $\{a_n\}$ in terms of $\alpha, \beta$.
b. If $p = 1$, $q = \frac{1}{4}$, find the sum of the first $n$ terms of $\{a_n\}$.