Cue Column:

• What is an AGP?
• What is the standard technique?
• Can we sum infinite AGP?

Note Column: An AGP is a series where the kth term = (AP term) * (GP term): S = a + (a+d)r + (a+2d)$r^2$ + (a+3d)$r^3$ + ...

Standard technique (S - rS):

1. Write S = a + (a+d)r + (a+2d)$r^2$ + ...
2. Write rS = ar + (a+d)$r^2$ + (a+2d)$r^3$ + ...
3. Subtract: S(1-r) = a + d(r + $r^2$ + $r^3$ + ...) = a + $\frac{dr}{1-r}$ (for infinite)
4. S(1-r) = a + $\frac{dr}{1-r}$
5. S = $\frac{a}{1-r}$ + $\frac{dr}{1-r}$^2

For finite AGP (n terms): S(1-r) = a + d(r + $r^2$ + ... + $r^{n-1}$) - [a+(n-1)d]$r^n$ = a + dr$\frac{1-r^{n-1}}{(1-r)}$ - [a+(n-1)d]$r^n$

Summary: AGP = multiply by r, subtract, simplify. The AP part telescopes to a constant difference, leaving a GP to sum.