Arithmetic sequence term is one of 4 sequences formulas in the algebra section of this library, and it is used at high school level.
Why arithmetic sequence term works
Getting from the first term to the nth term means taking steps of size d, and there is always one fewer step than there are terms. Term four is only three steps past term one. The n minus 1 is counting gaps between terms, not the terms themselves.
What each symbol means
$a_1$ is the first term, $d$ the common difference, and $n$ the term number.
Arithmetic sequence term: when it holds
$n$ is a positive integer and the difference must be constant.
When it stops applying
The gaps must all be the same size. In 2, 4, 8, 16 the gaps are 2, 4 and 8, so the sequence is not arithmetic and this formula would predict 2 plus 3 times 2, which is 8, where the real fourth term is 16.
Arithmetic sequence term: a worked example
If $a_1=4$ and $d=3$, then $a_{10}=4+9(3)=31$.
The mistake to avoid
What people do: Multiplying d by n instead of n minus 1, giving 34 for a sequence starting at 4 with difference 3.
Why it goes wrong: That takes one step too many. The tenth term is 4 plus nine steps of 3, which is 31, and the off-by-one error grows into every later answer built on it.
Do this instead: Count the steps on your fingers for a small case first. From term 1 to term 4 you move three times, which fixes the pattern as n minus 1.
Arithmetic sequence term: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a_1$ is the first term, $d$ the common difference, and $n$ the term number.
- Check the conditions before substituting. $n$ is a positive integer and the difference must be constant.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Sequences
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where arithmetic sequence term comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Sequences and Series — the lesson behind this formula: recognize arithmetic, geometric, and recursive patterns.
- Algebra Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about arithmetic sequence term
How do I find d when I am given two terms that are not next to each other?
Divide the difference of the values by the difference of the positions. With the third term 7 and the eighth term 22, that is 15 divided by 5, so d is 3 and the first term is 1.
Can the common difference be negative or fractional?
Yes to both. A negative d makes the terms shrink, and a fractional d is common in problems about steady measurements such as 0.25 centimetres of growth each week.
Is this the same thing as a linear function?
It has the same shape, with d acting as the slope and the graph forming evenly spaced dots on a line. The difference is that only whole-number positions exist, so there is no term between term 3 and term 4.
How do I tell whether a given value appears in the sequence?
Solve for n and check it is a positive whole number. For 4, 7, 10 and so on, the value 100 gives (100 - 4)/3 + 1 = 33, so 100 really is the 33rd term.