Arithmetic Sequences

Finding the common difference

Subtract any term from the term right after it: $d = a_2 - a_1$. To be safe, check a second pair — in a true arithmetic sequence, every gap is identical. If the gaps change, the sequence is not arithmetic and the formula does not apply.

Keep the order straight: later term minus earlier term. For $10, 7, 4, \ldots$, the difference is $7 - 10 = -3$, not $3$. The sign of $d$ tells you whether the sequence climbs or falls.

Why the formula uses $n - 1$

To get from the 1st term to the 5th term, you add $d$ four times — one step fewer than the term number, because the first term takes zero steps. In general, reaching term $n$ takes $n - 1$ steps of size $d$, which is exactly what $a_n = a_1 + (n - 1)d$ says.

Writing $a_n = a_1 + nd$ is the classic mistake. Test any formula on the first term: $a_1$ should come back unchanged, and only the $(n - 1)$ version does that.

Arithmetic sequences are linear functions in disguise

Plot the term number $n$ against the term value $a_n$ and the points line up on a straight line with slope $d$. A constant difference in a sequence is the same idea as a constant rate of change in a table. The only difference is the domain: $n$ only takes the values $1, 2, 3, \ldots$, so the graph is a row of separate points rather than a solid line.

Below, the sequence $3, 7, 11, 15, 19$ is plotted as separate points. They fall exactly on a line of slope $d = 4$ (shown dashed) — the sequence is a linear function restricted to whole-number inputs.

Worked examples

Try one yourself

Common questions