10.3 The nth term of a linear sequence — Learn

Prerequisites: times tables, substitution, and position-to-term rules.

Objectives: find an expression for the nth term of a linear sequence and use it to find a specified term. This section covers A25.

The common difference

A linear sequence has the same difference between consecutive terms. This difference is called the common difference.

For $7,11,15,19,\ldots$, the common difference is $4$. The coefficient of $n$ in its rule is $4$, so start with $4n$.

Example 1. Find the nth term of $7,11,15,19,\ldots$.

The $4n$ sequence is $4,8,12,16,\ldots$. Each given term is $3$ more.

Check position $1$: $4(1)+3=7$. Check position $2$: $4(2)+3=11$.

A decreasing sequence

The common difference can be negative.

Example 2. Find the nth term of $12,9,6,3,\ldots$.

The common difference is $-3$, so start with $-3n$.

At position $1$, $-3(1)=-3$. Add $15$ to reach the first term, $12$.

Check position $2$: $-3(2)+15=9$.

Progress check. Pattern A has $5,8,11,14,\ldots$. Its nth term is $3n+2$, so design 5 contains $3(5)+2=17$ tiles.

Watch out. Using the first term as the constant often gives the wrong rule. First match the common difference, then adjust the constant and check positions $1$ and $2$.

You should now be able to

• find the common difference in a linear sequence;
• use that difference as the coefficient of $n$;
• adjust the constant to match the first term;
• check the rule by substituting term positions.