Arithmetic Series Calculator
Sum of Series (Sₙ)25
nth Term (aₙ)9
Terms1, 3, 5, 7, 9
The Arithmetic Series Calculator takes a first term, a fixed common difference, and a term count, then returns both the value of the nth term and the running total of every term added together. An arithmetic sequence grows by adding the same amount at each step, so its partial sum follows a clean closed-form rule rather than requiring you to add each term by hand.
Formula
aₙ = a₁ + (n − 1)d ; Sₙ = n/2 × (a₁ + aₙ)
- a₁
- First term of the sequence
- d
- Common difference between consecutive terms
- n
- Number of terms being summed
- aₙ
- Value of the nth (last) term
- Sₙ
- Sum of the first n terms
How it works
- Enter the first term a₁, the common difference d (the constant amount added between consecutive terms, which may be negative), and the number of terms n you want to include.
- The calculator finds the nth term with aₙ = a₁ + (n − 1)d, then sums the run using Sₙ = n/2 × (a₁ + aₙ).
- It also lists the individual terms (up to the first 50) so you can verify the progression visually.
Worked example
Sum the first 5 terms of the sequence starting at 1 with a common difference of 2.
- nth term: a₅ = 1 + (5 − 1) × 2 = 1 + 8 = 9.
- Sum: S₅ = 5/2 × (1 + 9) = 2.5 × 10 = 25.
- The terms are 1, 3, 5, 7, 9.
The 5th term is 9 and the sum of the series is 25.
Frequently asked questions
- What is the difference between an arithmetic sequence and an arithmetic series?
- A sequence is the ordered list of terms (1, 3, 5, 7, …), while a series is the sum of those terms. This tool reports both the nth term of the sequence and the series total.
- Can the common difference be negative or zero?
- Yes. A negative difference produces a decreasing sequence, and a difference of zero produces a constant sequence where every term equals the first term, so the sum is simply a₁ multiplied by n.
- How does this differ from the number sequence calculator?
- The number sequence calculator inspects a list of numbers you already have and detects whether a pattern exists. This calculator instead builds a series from parameters you supply (a₁, d, and n) and computes its term and sum directly.
- Why are only the first 50 terms shown?
- The term list is capped at 50 entries to keep the display readable, but the nth term and the full sum are always computed for the complete count of terms you request.