Understanding Geometric Sequences

What is a Geometric Sequence?

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a fixed, non-zero number. This fixed number is called the common ratio, denoted by $r$.

For example, in the sequence $2, 6, 18, 54, \ldots$, each number is multiplied by $3$ to get the next one. Therefore, the common ratio is $r = 3$.

The Formula for the $n$th Term

To find any term in a geometric sequence without writing out the whole list, you can use the $n$th term formula:

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

• $a_n$ is the $n$th term you want to find.
• $a_1$ is the first term in the sequence.
• $r$ is the common ratio.
• $n$ is the position of the term.

Because the formula involves an exponent ($n-1$), geometric sequences are closely related to exponential functions. They grow or decay exponentially depending on whether the common ratio is greater than or less than $1$.

Example Problems

Example 1: Finding a specific term Find the 8th term of the sequence $2, 6, 18, 54, \ldots$

Solution:

1. Identify the first term: $a_1 = 2$.
2. Find the common ratio: $r = 6 / 2 = 3$.
3. Plug these into the formula for $n = 8$: $a_8 = 2 \cdot 3^{8-1}$ $a_8 = 2 \cdot 3^7$ $a_8 = 2 \cdot 2187 = 4374$

The 8th term is $4374$.

Example 2: Writing a formula Write a formula for the geometric sequence where $a_1 = 100$ and $r = 0.5$.

Solution: Substitute the given values directly into the general formula: $a_n = 100 \cdot (0.5)^{n-1}$

This formula can now be used to find any term in this specific sequence.