The question asks to prove that:
$S(n) = pr^{n-1} + q(n − 1)r^{n−1}$
is the closed form solution for linear, second order, constant coefficient, homogeneous recurrence relations where the characteristic equation:
$t^2 − c(1)t − c(2) = 0$
has a single root.
I believe the way to go about this problem is through the expand, guess, verify, guess equation? I'm saying that because that's what we've learned so far in class, but I'm just confused about how EGF would actually work out with this one.
As always, I'm not looking for answers, but simply hints on where to go from here and any hints would definitely be helpful as I want to learn as I go. Thank you so much
Hint
One root means
$$S_n=2cS_{n-1}-c^2S_{n-2}$$
with $c$ as the unique root.
thus
$$S_n-cS_{n-1}=c(S_{n-1}-cS_{n-2})$$
$\implies$
$$S_n=cS_{n-1}+c^{n-1}(S_1-cS_0)$$
$$S_{n-1}=cS_{n-2}+c^{n-2}(S_1-cS_0)$$
$$S_2=cS_1+c(S_1-cS_0)$$
finally by telescopage,
$$S_n=c^nS_1+nc^n(S_1-cS_0)$$