Key Differences Between Arithmetic and Geometric Sequences

Arithmetic and geometric sequences are two important types of sequences that you learn about in Algebra I. They each have their own unique features. Let’s break it down:

1. What They Are:

   • Arithmetic Sequence: This is a list of numbers where the difference between each number and the next one is the same. This difference is called $d$.
   • Geometric Sequence: In a geometric sequence, the numbers are related by a constant ratio, which we call $r$. This means you multiply by $r$ to get from one number to the next.

2. How to Write Them:

   • For an arithmetic sequence, you can find the $n$-th term (which means the term in that position) with this formula:

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

     Here, $a_1$ is the first number in the sequence, and $n$ is the position of the term.

   • For a geometric sequence, you can find the $n$-th term using this formula:

     $$g_n = g_1 \cdot r^{(n - 1)}$$

     In this case, $g_1$ is the first number, and $r$ is the ratio you multiply by.

3. Examples:

   • Arithmetic Sequence Example: Consider the sequence 2, 5, 8, 11, ... Here, the difference $d$ is 3, since you add 3 each time.

   • Geometric Sequence Example: For the sequence 3, 6, 12, 24, ... the ratio $r$ is 2, because you multiply by 2 to get to the next number.

4. Adding Up the First $n$ Terms:

   • To find the sum of the first $n$ terms of an arithmetic sequence (called $S_n$), you can use this formula:

     $$S_n = \frac{n}{2} (a_1 + a_n)$$

   • For the sum of the first $n$ terms of a geometric sequence, you use:

     $$S_n = g_1 \frac{1 - r^n}{1 - r} \quad \text{if } r \neq 1$$

5. Where We Use Them:

   • We use arithmetic sequences in situations where things increase by the same amount, like when you get a raise at work.

   • Geometric sequences appear when things grow or shrink quickly, such as in changes in population.

Knowing these key differences helps us see where to use each type of sequence in math and in everyday life.