An arithmetico-geometric sequence is the result of element-by-element multiplication of the elements of a geometric progression with the corresponding elements of an arithmetic progression.
Let \(A_i\) be the element of the \(i\)th arithmetic progression (starting from \(i=0\)), and let \(G_i\) be the value of the \(i\)th geometric progression (starting from \(i=0\)). So we can represent the series as:
$$S = A_0G_0 + A_1G_1 + A_2G_2 + \ldots + A_{n-1}G_{n-1}$$
There are \(n\) terms in total. Let \(A_i = a + id\) and let \(G_i = br^i\):
$$S = ab + (a + d)br + (a + 2d)br^2 + (a + 3d)br^3 + \ldots + (a + (n-1)d)br^{n-1}$$
We can write the product \(A_i G_i\) as:
$$A_i G_i = (a + id)br^i = abr^i + idbr^i$$
The sum of the series is:
$$ S = \sum_{i=0}^{n-1} abr^i + idbr^i = ab \sum_{i=0}^{n-1} r^i + db\sum_{i=0}^{n-1} ir^i $$
The first sum is a geometric series starting from 0 and ending with \(r^{n-1}\):
$$ S = ab \left( \frac{1-r^n}{1-r} \right) + db\sum_{i=0}^{n-1} ir^i $$
The second sum is already derived here:
$$ S = ab \left( \frac{1-r^n}{1-r} \right) + db \left( \frac{r(1-r^{n-1})}{(1-r)^2} - \frac{(n-1)r^n}{1-r}\right) $$
Expanding:
$$ S = \frac{ab- abr^n}{1-r} + \frac{dbr(1-r^{n-1})}{(1-r)^2} - \frac{dbnr^n - dbr^n}{1-r} $$
Rearranging:
$$\begin{align} S &= \frac{ab- abr^n + dbr^n - dbnr^n}{1-r} + \frac{dbr(1-r^{n-1})}{(1-r)^2} \\ &= \frac{ab - abr^n - dbnr^n}{1-r} + \frac{dbr^n}{1-r} + \frac{dbr(1-r^{n-1})}{(1-r)^2} \end{align}$$
Multiplying the seond term with \(\frac{1-r}{1-r}\):
$$\begin{align} S &= \frac{ab - abr^n - dbnr^n}{1-r} + \frac{dbr^n - dbr^{n+1}}{(1-r)^2} + \frac{dbr(1-r^{n-1})}{(1-r)^2} \\ &= \frac{ab - abr^n - dbnr^n}{1-r} + \frac{dbr^n - dbr^{n+1}+ dbr-dbr^n}{(1-r)^2} \end{align}$$
Simplifying:
$$\begin{align} S &= \frac{ab - (a + nd)(br^n)}{1-r} + \frac{dbr- dbr^{n+1}}{(1-r)^2} \\ &= \frac{ab - br^n(a + nd)}{1-r} + \frac{dr (b- br^n)}{(1-r)^2} \end{align}$$
Since \([A_0 = a]\), \([G_0 = b]\), \([A_n = a + nd]\) and \([G_n = br^n]\):
$$\begin{align} S &= \frac{A_0 G_0 - A_nG_n}{1-r} + \frac{dr}{(1-r)^2} (G_0 - G_n) \end{align}$$