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 + dr/(1-r) (for infinite)
4. S(1-r) = a + dr/(1-r)
5. S = a/(1-r) + 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(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.