When you start learning about sequences and series, you will mainly see two types: arithmetic and geometric sequences. It’s important to know how they are different to help you with pre-calculus.

1. What They Are:

• Arithmetic Sequence: This is a list of numbers where you get each number by adding the same amount (called the common difference, $d$) to the number before it. For example, in the list 2, 5, 8, 11, you add 3 each time, so here, $d$ equals 3.

• Geometric Sequence: This type is a little different. In a geometric sequence, you get each number by multiplying the number before it by the same amount (called the common ratio, $r$). For example, in the list 3, 6, 12, 24, you multiply by 2 each time, so here, $r$ equals 2.

2. How to Write Them:

• For an arithmetic sequence, you can write the formula like this:

  $a_n = a_1 + (n - 1)d$

  Here, $a_1$ is the first number and $d$ is the common difference.

• For a geometric sequence, the formula looks like this:

  $a_n = a_1 \cdot r^{(n - 1)}$

  Here, $a_1$ is the first number and $r$ is the common ratio.

3. How They Grow:

• In arithmetic sequences, the growth is straight and steady. This means the numbers go up at the same rate. If you keep going with the earlier example (2, 5, 8, 11), you will keep adding 3 each time.

• On the flip side, geometric sequences grow really fast. For example, in the list 3, 6, 12, 24, each number is double the one before. This means they get much bigger very quickly!

By knowing these differences, you can use each type of sequence in different math situations!