10.1 Term-to-term and position-to-term rules — Learn

Prerequisites: addition, subtraction, multiplication, and substitution into an expression.

Objectives: describe how terms change and use a position number to find a term directly. This section covers A23.

Term-to-term rules

A term is one number in a sequence. A term-to-term rule tells you how to get from one term to the next.

For $4, 7, 10, 13,\ldots$, add $3$ each time. The dots mean that the pattern continues.

Example 1. Write the next two terms.

The next terms are $16$ and $19$.

Position-to-term rules

The position is a term's place in the list. A position-to-term rule uses the position number to find a term without listing every earlier term.

In $3,6,9,12,\ldots$, each term is $3$ times its position. So the rule is $3n$, where $n$ is the position.

Example 2. Find the 20th term.

The 20th term is $60$.

For $5,8,11,14,\ldots$, each term is $3$ times its position, then $2$ more. The rule is $3n+2$. At position $4$:

Progress check. Pattern A in the chapter problem follows $3n+2$. Its position 5 term is $3(5)+2=17$ tiles.

Watch out. The first term is not always the change between terms. For $5,8,11,\ldots$, the first term is $5$ and the change is $3$. Check a rule by substituting positions $1$ and $2$.

You should now be able to

• state the term-to-term change in a sequence;
• continue a sequence using its rule;
• calculate a term from its position number;
• check a position-to-term rule against the first terms.