Arithmetic Series Calculator
Arithmetic series: what this calculator finds
An arithmetic sequence is a list of numbers where each term changes by a constant amount called the common difference (d). An arithmetic series is the sum of the first n terms of that sequence.
With inputs first term (a1), common difference (d), and number of terms (n), this calculator computes:
- The nth (last) term: an
- The sum of the first n terms: Sn
How to use this arithmetic series calculator
- Enter the first term a1 (any real number, including negatives and decimals).
- Enter the common difference d (can be negative, zero, or decimal).
- Enter the number of terms n (typically a whole number: 1, 2, 3, …).
- Click Calculate to get an and Sn.
Definitions
- a1: first term of the sequence
- d: common difference (the amount added each step)
- n: number of terms being included
- an: nth term (also the last term among the first n)
- Sn: sum of the first n terms
Formulas
Nth term of an arithmetic sequence:
an = a1 + (n − 1)d
Sum of the first n terms (two equivalent forms):
Sn = (n/2)·(2a1 + (n − 1)d)
Sn = (n/2)·(a1 + an)
MathML (copy/paste friendly)
Interpreting the results
What does an mean?
an is the value of the term at position n. If you list the first n terms, an is the last one shown.
What does Sn mean?
Sn is the total when you add the first n terms:
a1 + a2 + … + an
Special cases
- If d = 0: every term equals a1, so Sn = n·a1.
- If d < 0: the sequence decreases, but the formulas still work normally.
- If a1 or d are decimals: results may be fractional; that is expected.
Worked example (step by step)
Suppose a1 = 2, d = 3, and n = 4.
1) Find the 4th term
a4 = 2 + (4 − 1)·3 = 2 + 9 = 11
2) Find the sum of the first 4 terms
S4 = (4/2)·(2·2 + (4 − 1)·3) = 2·(4 + 9) = 26
Check by listing the terms: 2, 5, 8, 11 → 2 + 5 + 8 + 11 = 26.
Sequence vs. series (quick comparison)
| Concept | What it represents | Typical notation | Example (a₁ = 2, d = 3) |
|---|---|---|---|
| Arithmetic sequence | The list of terms | aₙ | 2, 5, 8, 11, … |
| Arithmetic series | The sum of the first n terms | Sₙ | S₄ = 2 + 5 + 8 + 11 = 26 |
More examples (common classroom cases)
- Decreasing sequence: a₁ = 20, d = −2, n = 6 → a₆ = 20 + 5(−2) = 10; S₆ = (6/2)(20 + 10) = 90.
- Constant sequence: a₁ = 7, d = 0, n = 10 → a₁₀ = 7; S₁₀ = 10·7 = 70.
- Decimal difference: a₁ = 1.5, d = 0.5, n = 8 → a₈ = 1.5 + 7(0.5) = 5; S₈ = (8/2)(1.5 + 5) = 26.
Limitations and assumptions
- n should be a positive integer. If you enter a non-integer n, the formulas still evaluate algebraically, but it may not represent a sum of a whole number of terms.
- Applies only to arithmetic sequences (constant difference). If differences vary, these formulas do not apply.
- Floating-point rounding: JavaScript uses floating-point arithmetic, so very large values or certain decimals can show rounding (e.g., 0.1 + 0.2 effects).
- Very large inputs: extremely large n or large magnitudes of a1/d can exceed safe integer precision; consider a big-number tool if you need exact integer results at huge scales.
FAQ
Can the common difference d be negative?
Yes. A negative d means the sequence decreases by a fixed amount each term, and the same nth-term and sum formulas still apply.
What if d = 0?
Then every term equals a1. The sum becomes Sn = n·a1.
Is an arithmetic series the same as an arithmetic sequence?
No. A sequence is the list of terms (an), while a series is the sum of terms (Sn).
Can a₁ or d be decimals?
Yes. The results can be fractional/decimal as well.
Why does the sum formula use n/2?
The series can be paired from the beginning and end: each pair sums to the same value (a1 + an), and there are n/2 such pairs (with a middle term when n is odd).
Can this calculator find n if I know the sum?
Not directly. Solving for n from Sn typically leads to a quadratic equation when d ≠ 0.