Arithmetic Series Sum Calculator
Calculate the sum of the first n terms of an arithmetic series from the first term and common difference, or from the first and last terms.
The arithmetic series sum equals the average of the first and last terms multiplied by the number of terms. This is the same pairing idea used for the sum of the numbers from 1 to 100.
How to calculate an arithmetic series sum
This calculator is for the sum of the first n terms of an arithmetic sequence. It supports the two common school inputs: first term with common difference, or first term with last term.
S_n is the sum of the first n terms, n is the number of terms, a_1 is the first term, and a_n is the last term among the first n terms.
S_n is the sum of the first n terms, n is the number of terms, a_1 is the first term, and d is the common difference.
a_n is the last term among the first n terms, a_1 is the first term, n is the number of terms, and d is the common difference.
Which inputs to use
| Known values | Calculator output | Best for |
|---|---|---|
| a1, d, and n | Finds the last term and the sum of the first n terms | Problems that give a common difference |
| a1, an, and n | Finds the common difference and the sum | Problems that give the first and last terms |
| other unknowns | Not solved on this focused page | Use a broader sequence solver if another parameter is missing |
- n must be a positive integer.
- When n equals 1, the sum is the first term.
- A negative or zero common difference is valid.
- Very large inputs may exceed JavaScript safe integer precision.
Examples
| Given | Check | Sum |
|---|---|---|
| ,, | sum of the numbers from 1 to 100 | 5050 |
| ,, | positive common difference with last term 29 | 155 |
| ,, | negative common difference with last term 0 | 30 |
| ,.5, | decimal common difference with last term 10.5 | 115 |
Calculator limits
- The calculator sums the first n terms, not an arbitrary middle slice of a sequence.
- It does not solve for n from a known sum.
- It does not list every term or work as a symbolic algebra solver.
- Financial examples with evenly changing payments are only analogies; this is not a loan calculator.
- If several sequence parameters are unknown, use a broader arithmetic sequence tool.
| Scenario | Use |
|---|---|
| Need one term only | Arithmetic sequence nth term calculator |
| Need several unknown parameters | General arithmetic sequence calculator |
| Sequence changes by multiplication | Geometric sequence calculator |
Frequently Asked Questions
Sources and References
Calculations are based on the listed reference sources. Links open in a new tab.
Related Tools
Find the nth term of an arithmetic sequence from the first term, common difference, and term number. The calculator also shows the substitution and a short sequence check.
Calculate an arithmetic sequence from the first term and common difference, or from the first two terms. See the nth term, the sum of the first terms, and the generated sequence.
Calculate a geometric sequence from the first term and common ratio, or from the first two terms. See the nth term, first n-term sum, and infinite sum when it converges.