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 + ... )?
Click here to download the GP Algebraically worksheet.
Click here for the Notes for Teachers.
Click here to download the GP Algebraically poster.
If you enjoyed this article, please consider sharing it!
FOLLOW US ON SOCIAL MEDIA
