Arithmetic Sequence Calculator
Nth term, partial sum, and explicit rule from first term and difference.
What it does
Arithmetic sequences march by constant steps: each term adds the same difference to the last. The nth term multiplies that step by n minus one onto the first, and the partial sum averages first with last times count.
The tool above takes first term with difference and term number. It returns the nth term with the partial sum and the explicit rule beside them.
The default of 7 with difference 3 at n = 100 gives a100 = 304 with S100 = 15,550 and rule an = 3n + 4. A falling 6 with difference -2 at 75 gives a75 = -142.
The formulas follow open algebra textbook treatments with matching worked examples. Both write an as a1 plus (n-1)d and Sn as n times first-plus-last over two.
How to use
- Enter the first term a1.
- Enter the common difference d, signed.
- Enter which term number n to find.
- Read the nth term first.
- Read the sum for the running total.
Formula
- d
- Constant step between successive terms.
- n
- One based term position.
- Sn
- Sum of the first n terms.
Constant difference defines arithmetic: doubling differences mean geometric instead. Negative differences march downward with the same formulas.
Worked example
Example: 7 with difference 3, n = 100
Defaults: a1 7, d 3, n 100.
- Nth: 7 + (100-1) x 3 = 7 + 297 = 304.
- Sum: 100 x (7 + 304) / 2 = 15,550.
- Rule: an = 3n + (7-3) = 3n + 4.
- Check: n=1 gives 7, n=2 gives 10.
- Tool shows a100 = 304, S100 = 15,550, an = 3n + 4.
Reference table
| First / d / n | Nth term | Sum |
|---|---|---|
| 7 / 3 / 100 | a100 = 304 | S100 = 15,550 |
| 6 / -2 / 75 | a75 = -142 | S75 = -5,100 |
| 2 / 12 / 5 | a5 = 50 | S5 = 130 |
Limits
- Constant difference only; curves need other families.
- Huge n values overflow display past float range.
- Sums assume exact terms, not rounded lists.
- Infinite sums diverge unless trivially zero.
Common mistakes
- Multiplying d by n instead of n-1.
- Sign errors on negative differences.
- Averaging wrong endpoints for the sum.
- Calling geometric doubling arithmetic.
- Starting counts at zero against the formula.
Try it yourself
Quick checks. Answers sit tucked under each question.
6, 4, 2... 75th term?
d = -2: a75 = 6 + 74 x -2 = -142, sum -5,100.
First five of an = 3n + 4?
n=1..5 gives 7, 10, 13, 16, 19.
Is 1, 2, 4, 8 arithmetic?
No: differences 1, 2, 4 grow geometrically instead.
Questions
What is an arithmetic sequence?
Constant steps between terms: 7, 10, 13 with difference 3.
Nth term formula?
a1 + (n-1)d: first term plus steps times gaps before it.
Sum formula?
n times first-plus-last over two: pairs ends inward.
Negative differences?
Same formulas marching down: 6 with -2 reaches -142 at 75.
Arithmetic versus geometric?
Constant difference versus constant ratio: plus 3 versus times 2.
What is the explicit rule?
an as dn plus a constant: 3n + 4 here.