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:

$$\begin{align} & 4,\ 8,\ 16,\ 32,\ 64,\ \ldots \\ & 1,\ 3,\ 4,\ 6,\ 7,\ \ldots \\ & 3,\ 5,\ 7,\ 9,\ 11,\ \ldots \end{align}$$Next, give your students these challenges:

- Sort Them Out (1) by NRICH
- 2000 AMC 8, Problem 6
- 2020 AMC 8, Problem 5
- 2000 AMC 8, Problem 7
- Featuring Factorials! by Pierce School: Problem / Solution

George and Jim want to buy a chocolate bar. George would need 2 cents more to buy it. Jim would need 50 cents more. When they put their money together, it's still not enough to buy the chocolate bar. How much does it cost?

Here's one solution: Let \(c\) be the cost of the chocolate bar, \(g\) the amount of money George has, \(j\) the amount Jim has. Then we have \(g = c - 2\) and \(j = c - 50.\) We know their money combined isn't enough to buy the chocolate bar, so \((c - 2) + (c - 50) \lt c.\) Solving the inequality, we find \(c \lt 52\) cents. But \(g = c - 50\) and \(g \ge 0,\) so \(c \ge 50.\) Thus, the cost of the chocolate bar is either 50, or 51 cents. Another, albeit, less favorable solution, is to use guess and check, as seen here.

Note: The problem comes from here. I have adapted it for Americans, rephrased the question slightly, and succinctly explained the solution.

Conclude by leading this investigation:

Kajitsu – symmetry puzzles

by MathPickle