I have to solve the following recurrence, given $T(1)=1$, $$T(n)=aT(n-1)+bn$$
I have done the following:
$$T(n)=aT(n-1)+bn \\ =a^2T(n-2)+ab(n-1)+bn \\ =a^3T(n-3)+a^2b(n-2)+ab(n-1)+bn \\ = \dots \\ =a^iT(n-i)+b\sum_{k=0}^{i-1} a^k (n-k)$$
$n-i=1 \Rightarrow i=n-1$
$$T(n)=a^{n-1}T(1)+b \sum_{k=0}^{n-2}a^k (n-k)=a^{n-1}+bn\sum_{k=0}^{n-2}a^k-b \sum_{k=0}^{n-2}k a^k \\ =a^{n-1}+bn \frac{a^{n-1}-1}{a-1}-b \frac{(n-2)a^n-(n-1)a^{n-1}+a}{(a-1)^2}$$
Could you tell me if my result is correct?