Introduction to Sequences: Geometric Sequences vs Other Types of Sequences

What Are Sequences?
A sequence is a list of numbers arranged in a specific order. Each number in the list is called a "term".

There are different types of sequences, including:

• Arithmetic sequences
• Geometric sequences
• Other types that don't fit neatly into these categories

What Are Geometric Sequences?
A geometric sequence is special because each term is found by multiplying the previous term by the same number. This number is called the "common ratio".

For example, if the first term is $a$ and the common ratio is $r$, we can find the $n$-th term using this formula:

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

Here’s a quick example:
If we start with 2 and multiply by 3 each time, we get:
2, 6, 18, 54, … (In this case, $r = 3$)

Characteristics of Geometric Sequences:

• These sequences can grow or shrink quickly because of their multiplying nature.
• The ratio $r$ can be any real number, as long as it is not zero.

How Geometric Sequences Compare to Other Sequences:

1. Arithmetic Sequences:

   • In an arithmetic sequence, we add or subtract a constant amount.
   • The general formula looks like this:
     $a_n = a + (n-1)d$
     where $d$ is the common difference.

2. Other Types of Sequences:

   • These can be very different and may not follow a consistent pattern.
   • For example, in the Fibonacci sequence, each number is the sum of the two before it.

Conclusion:
Geometric sequences stand out because they use multiplication to find new terms. In contrast, arithmetic sequences rely on addition, while other types of sequences can have a variety of patterns.