Unit 1, Lesson 2
Introducing Geometric Sequences
Algebra 2
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Clare takes a piece of paper, cuts it in half, then stacks the pieces. She takes the stack of two pieces, cuts it in half again to form four pieces, and then stacks them again. She keeps repeating the process.
| number of cuts |
number of pieces |
area, in square inches, of each piece |
|---|---|---|
| 0 | ||
| 1 | ||
| 2 | ||
| 3 | ||
| 4 | ||
| 5 |
Consider the sequence 2, 6, 18, . . . How would you describe how to calculate the next term from the previous?
In this case, each term in this sequence is 3 times the term before it.
Here is a way to describe this sequence: the starting term is 2, and the
This is an example of a geometric sequence. A geometric sequence is a sequence in which the value of each term is the value of the previous term multiplied by a constant. If we know the constant to multiply by, we can use it to find the value of other terms.
This constant multiplier (the “3” in the example) is often called the sequence’s growth factor or common ratio. One way to find it is to divide consecutive terms. This can also help us decide whether a sequence is geometric.
The sequence 1, 3, 5, 7, 9 is not a geometric sequence because
The sequence 100, 20, 4, 0.8, however, is geometric because if we divide each term by the previous term, we get 0.2 each time:
A sequence in which each term is a constant times the previous term.