Geometric sequences and arithmetic sequences are two important types of sequences we learn about in math.

What They Are:

• Arithmetic Sequence: This is a list of numbers where you get each number by adding the same amount each time. This amount is called the common difference ($d$).

  You can find the nth term like this:
  $a_n = a_1 + (n - 1)d$

  For example:
  In the sequence 2, 5, 8, 11, you add 3 each time. So here, the common difference $d$ is 3.

• Geometric Sequence: In this type, you find each number by multiplying the previous one by a constant number, known as the common ratio ($r$).

  Here’s how to find the nth term:
  $a_n = a_1 \cdot r^{(n-1)}$

  For example:
  In the sequence 3, 6, 12, 24, you multiply by 2 each time. So, the common ratio $r$ is 2.

Main Differences:

1. How They Grow:

   • Arithmetic: The numbers grow in a straight line because you are adding.
   • Geometric: The numbers grow faster because you are multiplying.

2. Examples:

   • Arithmetic: For the sequence 4, 7, 10, you add 3 each time (common difference of 3).
   • Geometric: For the sequence 5, 15, 45, you multiply by 3 each time (common ratio of 3).

3. Where We Use Them:

   • We often use arithmetic sequences in budgeting or planning money.
   • Geometric sequences are useful for understanding things like populations, investments, and technology growth.

By learning about these sequences, we can better understand patterns and make sense of the world around us!