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Part 1: Finding a closed-form expression for a sum

Simplify the following products.
$(1-r)(1+r) =$
$(1-r)(1+r+r^2) =$
$(1-r)(1+r+r^2+r^3) =$
$(1-r)(1+r+r^2+r^3+ \cdots + r^n) =$

Assume $r\neq 1$ . Using what you calculated above, find a closed-form expression for the following finite sum.
$1+r+r^2+r^3+ \cdots + r^n =$

Part 2: What happens as n goes to infinity?

Part 3: Geometric series in general

Part 4: A specific geometric series