GP Algebraically

By toni On 26 May, 2017 · Leave a Comment

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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