An arithmetico-geometric series has the form: sum = a + (a+d)r + (a+2d)$r^2$ + ...

Infinite AGP (|r| < 1): S = $\frac{a}{1-r}$ + $\frac{dr}{1-r}$^2

Finite AGP (n terms): Step 1: Write S = a + (a+d)r + (a+2d)$r^2$ + ... + [a+(n-1)d]r^(n-1) Step 2: Multiply by r: rS = ar + (a+d)$r^2$ + ... + [a+(n-1)d]$r^n$ Step 3: Subtract: S(1-r) = a + d(r+$r^{2+}$...+r^(n-1)) - [a+(n-1)d]$r^n$ Step 4: Sum the GP: S(1-r) = a + dr(1-r^(n-1))/(1-r) - [a+(n-1)d]$r^n$ Step 5: Solve for S.

Common examples:

• sum k*$x^k$ = $\frac{x}{1-x}$^2 (for |x|<1)
• sum $k^2$*$x^k$ = x$\frac{1+x}{(1-x)}$^3
• 1+2x+3$x^{2+}$... = $\frac{1}{1-x}$^2