This number sequence calculator builds arithmetic terms from a first value, a common difference, and a count of terms. With first 2, diff 3, and n 8, the list is 2, 5, 8, 11, 14, 17, 20, 23.
Use it for homework checks, pattern practice, and quick term lists before sums or series lessons. For divisor style number work, open the factor calculator. For shared factors across integers, try the GCF calculator.
How the formula works
Each term equals first plus i times diff, for i = 0, 1, 2, through n-1. Each step adds the same difference. The last listed term uses i = n-1.
Worked example
first = 2, diff = 3, n = 8. Terms: 2, 2+3=5, 5+3=8, then 11, 14, 17, 20, and 23. Index check for the last term: 2 + 7*3 = 23.
| Input | Value |
|---|---|
| first | 2 |
| diff | 3 |
| n | 8 |
| kind | arithmetic |
| Sequence | 2, 5, 8, 11, 14, 17, 20, 23 |
How to use the fields
- Enter the first term.
- Enter the common difference.
- Enter how many terms to list, then read the generated sequence.
Zero based index reminder
The formula uses i starting at 0 so the first term is unchanged. People who memorize term n equals first plus (n-1) times diff with a 1 based n are saying the same thing.
Common mistakes
- Using i = n instead of i = n-1 for the last term
- Mixing geometric ratio rules into an arithmetic problem
- Changing the difference midway and still calling it arithmetic
- Counting terms incorrectly after editing n
Decreasing sequences
If diff is negative, each step subtracts. The algebra is identical. Only the sign of the difference changes the direction of the list.
Classroom warmups
Give students first 2 and diff 3 and ask for eight terms before they open the tool. Matching 2 through 23 confirms both addition fluency and indexing.
Then ask for only the fifth term: 2 + 4*3 = 14. Spot checks beat copying an entire list when time is short.
Link to sums later
Once terms are listed, arithmetic series formulas can sum them. This calculator focuses on generating the list so the inputs to a sum are visible and auditable.
Word problems
Seat numbers, evenly spaced posts, and savings that grow by a fixed dollar amount all map to arithmetic sequences. Translate the story into first, diff, and n before computing.
If a story multiplies instead of adds, switch to a geometric model. Forcing arithmetic on multiplicative growth produces wrong terms.
Checking by reverse difference
Subtract neighboring terms in the output. Every gap should equal diff. If one gap differs, a typing error or wrong n is likely.
For the default list, every step is +3 from 2 to 23, which is a fast verbal audit.
Decimals and fractions
The same rule accepts fractional first terms or differences. Keep consistent decimal precision so later terms do not drift from early rounding.
When a textbook uses fractions, convert carefully or keep fraction form on paper beside the calculator output.
Choosing n wisely
Large n makes long lists that are hard to paste into homework. Generate only the terms you need, or compute a single indexed term with first + (n-1)*diff when the full list is unnecessary.
Limitations
This page path emphasizes arithmetic sequences with a fixed difference. It is not a full recursive sequence solver for arbitrary custom rules.
From list to next term prediction
Once you know first and diff, the next term after the list is first + n*diff. For the default inputs that next term would be 2 + 8*3 = 26. Predicting beyond the printed list is the same rule with a larger index.
Teachers can hide the diff, show 2, 5, 8, 11, and ask students to recover diff = 3 before generating the remaining terms through 23.
Notation that stays clear in notes
Write first, diff, n, and the full term list on one line when you submit homework. Graders can then verify both the rule and the arithmetic without guessing which inputs produced 2, 5, 8, 11, 14, 17, 20, 23.
If you only need one distant term, skip listing everything and compute first + (n-1)*diff directly, then state n so the index convention is obvious.