Pre-Algebra
Number sequences
Arithmetic sequences
20 practice questions
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Theory + worked examples
Theory
An arithmetic sequence adds a constant common difference \(d\) from term to term.
The \(n\)th term is \(a_n=a_1+(n-1)d\).
Terms rise by a constant amount.
An arithmetic sequence.
The nth term:
\[a_n=a_1+(n-1)d\]
\(d\) is the constant difference between terms.
How to work with an arithmetic sequence
- Subtract consecutive terms to find \(d\).
- Add \(d\) to get the next term.
- Use \(a_n=a_1+(n-1)d\) for any term.
- Check with a known term.
Example 1 — Common difference
Find \(d\) for \(3,7,11,15,\ldots\)
Solution
Subtract consecutive terms.
| \(7-3\) | \(=\) | \(4\) |
Example 2 — Next term
Find the next term of \(3,7,11,15,\ldots\)
Solution
Add \(4\).
| \(15+4\) | \(=\) | \(19\) |
Example 3 — nth term
Find the \(10\)th term of \(3,7,11,\ldots\)
Solution
Use \(a_1+(n-1)d\).
| \(3+9\cdot4\) | \(=\) | \(39\) |
Example 4 — Write a rule
Write the rule for \(5,8,11,\ldots\)
Solution
\(a_1=5,\ d=3\).
| \(a_n\) | \(=\) | \(5+(n-1)3\) |
Common pitfalls
Use \((n-1)\), not \(n\), in the formula.
The difference is constant — check it.
Arithmetic adds; geometric multiplies.
Frequently asked questions
What is an arithmetic sequence?
One that adds a constant difference.
What is the common difference?
The constant added each term.
Next term of \(3,7,11,15\)?
\(19\).
What is the nth-term formula?
\(a_n=a_1+(n-1)d\).
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