Students will learn what a geometric series is, how to test whether a series is geometric, and how to extend a sequence which is known to be geometric. Students should see at least one example of a geometric sequence, such as:
$$1,\ 2,\ 4,\ 8,\ 16,\ \ldots$$
and at least one non-example of a non-geometric sequence, such as
$$1,\ 3,\ 6,\ 10,\ 15,\ \ldots$$
Example problems:
Determine whether each of the following series is arithmetic, geometric, or neither:
$$\begin{align}
& 4,\ 8,\ 16,\ 32,\ 64,\ \ldots \\
& 1,\ 3,\ 4,\ 6,\ 7,\ \ldots \\
& 3,\ 5,\ 7,\ 9,\ 11,\ \ldots
\end{align}$$
Concude by giving your students this challenge:
A Curious Surface Area by MAA: Give students the problem and allow them to find their own method of solution. Afterwards, show students how they could solve the problem mentally. That is, the front and back faces each add to 1/2, so together, that's 1. Each side face is equivalent to a side of the original cube, so they're each 1, and together, they're 2, so now we're at 3. There are 4 top faces, so there must also be 4 bottom faces, together, that's 8. Adding this to our previous total of 3, we get 11. This problem is easy to solve mentally because you're always keeping a whole number in your head while deducing the next addend.
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