Nth term and interpretation
The nth term is the value at the requested one-based position. The sum includes every term from the first through that position. The preview shows up to the first twelve terms.
To estimate a value between two known points, use the Linear Interpolation Calculator and follow its own input conventions.
The formula
Multiply the common difference by one less than the index, then add the first term. For the finite sum, multiply the term count by the average of the first and last terms.
Verified worked example
Starting at 3 and adding 2 each time gives 3, 5, 7, 9, and so on. Term 10 is 21. The first ten terms sum to 10 × (3 + 21) / 2 = 120.
How to use this calculator
- Enter the first term and the amount added at each step.
- Enter a whole term number starting at one.
- Read the nth term, the sum through that term, and the opening-term preview.
Arithmetic Sequence input conventions
An arithmetic progression increases or decreases by the same amount at every step. The index starts at one, so the first term has zero increments. A negative difference produces a descending sequence, and zero leaves every term unchanged. The preview shows up to twelve entries; it is a check on your chosen pattern rather than a complete list of a million terms.
Interpreting nth term
Pairing the first and last terms explains the sum: every opposite pair has the same total. An odd-length list has a middle term that equals half of a pair. This shortcut works for decimal and negative entries as well as integers. It does not identify a pattern from an arbitrary series, and a linearly changing quantity should not be confused with repeated percentage growth.
Assumptions & limitations
What this calculation assumes
- Every successive term differs by the supplied fixed amount, with the first term indexed as one.
What to keep in mind
- Finite real arithmetic progressions only; decimal results use floating-point arithmetic.
Common questions
Does the term number start at zero?
No. Enter 1 for the first term, 2 for the second, and so on. If your course uses a zero-indexed sequence, shift its index by one before entering it.
Can the difference be a decimal?
Yes. A fixed increment such as 0.25 is arithmetic. Keep the units consistent; the sum has the same unit as an individual term.
Can the sequence decrease?
Yes. Enter a negative common difference, such as −3. Each successive term is then three less than the preceding one; the sum still includes all terms through the selected index.
Sources & further reading
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