等差数列计算器
免费等差数列计算器。输入首项、公差和项数，得到第 n 项 aₙ 与数列之和 Sₙ。
根据首项和公差，求等差数列的第 n 项及其前 n 项之和。
Arithmetic Sequence Calculator
An arithmetic sequence (or arithmetic progression) is a list of numbers in which each term differs from the one before it by a fixed amount called the common difference, d. This calculator finds the nth term and the sum of the first n terms from the first term, the common difference, and how many terms you want.
What is an arithmetic sequence?
The sequence 2, 5, 8, 11, 14, … is arithmetic with first term a₁ = 2 and common difference d = 3 — each step adds 3. Two formulas describe it completely:
- nth term: aₙ = a₁ + (n − 1)·d
- Sum of n terms: Sₙ = n⁄2 · (a₁ + aₙ) = n⁄2 · (2a₁ + (n − 1)·d)
The sum formula is the one famously rediscovered by a young Gauss: to add 1 + 2 + … + 100, pair the ends (1 + 100 = 101) and note there are 50 such pairs, giving 5050.
How to use this calculator
- Enter the first term, a₁ — where the sequence starts.
- Enter the common difference, d — the constant step between terms (negative for a decreasing sequence).
- Enter the number of terms, n — how far into the sequence to go.
- Read the nth term and the sum.
Examples
Counting by threes. With a₁ = 2 and d = 3, the 10th term is 2 + 9 × 3 = 29, and the sum of the first 10 terms is 10⁄2 × (2 + 29) = 155.
Gauss’s sum. With a₁ = 1, d = 1, n = 100, the 100th term is 100 and the sum is 5050.
Decreasing. With a₁ = 10, d = −2, n = 5, the sequence is 10, 8, 6, 4, 2: the 5th term is 2 and the sum is 30.
Frequently asked questions
What is the common difference? It is the constant value added to move from one term to the next. Subtract any term from the one after it to find d.
Can the common difference be negative or zero? Yes. A negative d gives a decreasing sequence; a d of 0 repeats the first term.
What is the difference between a sequence and a series? A sequence is the list of terms; a series is the sum of those terms. The “sum” output here is the arithmetic series Sₙ.