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 \(-1<r<1 :\ \sum_{i=0}^{\infty} r^i = (1 +r + r^2 + r^3 + … )?\)

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