A sequence is an ordered list of numbers that follows a rule. Being able to describe the rule for the next term involves explaining the pattern in words and also using symbols where appropriate.

 

The rule should clearly show how to get from one term to the next.

 

 

Describing the Rule in Words

To describe a rule in words, state the operation that is applied each time.

 

For example, consider the sequence:

$$
4,\ 7,\ 10,\ 13
$$

 

The difference between each term is \( 3 \).

 

The rule in words is:

“Add 3 to get the next term.”

 

Another example:

$$
81,\ 27,\ 9,\ 3
$$

 

Each term is divided by \( 3 \).

 

The rule in words is:

“Divide by 3 to get the next term.”

 

Clear wording is important so the rule can be followed unambiguously.

 

 

Describing the Rule Using Symbols

The rule can also be written using symbols.

 

For an additive sequence, the rule can be written as:

$$
\text{next term} = \text{previous term} + 3
$$

 

For a multiplicative sequence, the rule can be written as:

$$
\text{next term} = \frac{\text{previous term}}{3}
$$

 

These symbolic rules show exactly how the next term is generated.

 

 

Rules Involving More Than One Operation

Some sequences use more than one operation.

 

For example:

$$
2,\ 6,\ 5,\ 15,\ 14
$$

 

The pattern alternates between multiplying by \( 3 \) and subtracting \( 1 \).

 

The rule in words is:

“Multiply by 3, then subtract 1, and repeat.”

 

The rule in symbols can be written as:

$$
\times 3,\ -1,\ \times 3,\ -1
$$

 

 

Describing the Next Term Explicitly

Sometimes the rule is used to state the actual next term.

 

For example, for the sequence:

$$
5,\ 10,\ 20
$$

 

The rule is multiply by \( 2 \), so the next term is:

$$
40
$$

 

The rule can be described as:

“Increase by a factor of 2 each time.”

 

 

Key Points to Remember

Identify how each term changes to the next.
Describe the rule clearly in words.
Use symbols to show the operation mathematically.
Make sure the rule works for every step in the sequence.

 

Being able to describe sequence rules accurately is essential for working with patterns, sequences and algebraic thinking.