Geometric Sequences and Common Ratios

A geometric sequence is an ordered list of numbers where each term is found by multiplying the previous term by the same fixed number. While arithmetic sequences grow by adding, geometric sequences grow (or shrink) by multiplying.

The Common Ratio ($r$)

The fixed number used to multiply is called the common ratio, represented by the letter $r$. To find the common ratio of a geometric sequence, simply divide any term by the term immediately before it.

For example, in the sequence $2, 6, 18, 54, \dots$: $r = \frac{6}{2} = 3$

The Geometric Sequence Formula

You don't have to multiply step-by-step to find a term far down the line. You can find any term in a geometric sequence using this 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.

Example Problems

Example 1: Find the common ratio and 6th term of the sequence $2, 6, 18, 54, \dots$

• First, find the common ratio: $r = \frac{6}{2} = 3$.
• The first term is $a_1 = 2$.
• To find the 6th term ($n = 6$), use the formula: $a_6 = 2 \cdot 3^{6-1} = 2 \cdot 3^5$ $a_6 = 2 \cdot 243 = 486$

Example 2: Is the sequence $1, 4, 16, 64$ arithmetic or geometric?

• Check for an arithmetic sequence (adding): $4 - 1 = 3$, but $16 - 4 = 12$. The difference is not constant.
• Check for a geometric sequence (multiplying): $\frac{4}{1} = 4$, and $\frac{16}{4} = 4$.
• Since the ratio is constant, the sequence is geometric with a common ratio of $r = 4$.

Example 3: Write the first 5 terms of a sequence where $a_1 = 3$ and $r = 2$.

• Start with the first term: $3$.
• Multiply by $2$ to get the next term: $3 \cdot 2 = 6$.
• Keep multiplying by $2$: $6 \cdot 2 = 12$, $12 \cdot 2 = 24$, $24 \cdot 2 = 48$.
• The first 5 terms are: $3, 6, 12, 24, 48$.