Multiply out these expressions:

$$(1 +r)(1 – r),\\ (1 +r + r^2)(1 – r),\\ (1 +r + r^2 + r^3)(1 – r),\\ (1 +r + r^2 + r^3 + r^4)(1 – r),\\ …$$

What do you notice?

Does this pattern continue?

Can you prove it?

What does this tell you about the sum to n terms of the geometric series with first term 1 and common ratio r?

Why do we get a formula for all values of r except r = 1?

Explain the formula: $$\sum_{i=0}^{n-1}r^i = 1 + r + r^2 + r^3 + … +r^{n-1} = \frac {1-r^n}{1-r}$$ for all $$r \neq 1.$$

What can you say about the limit of $$r^n$$ for $$-1<r<1$$ as $$r\to \infty$$?

What does this suggest to you about the infinite sum of the geometric series for $$-1<r<1 :\ \sum_{i=0}^{\infty} r^i = (1 +r + r^2 + r^3 + … )?$$