Mathematics, Statistics & Geometry

Arithmetic Sequence Calculator

Use this free Arithmetic Sequence Calculator to find any sequence term, the common difference, or the sum. Shows step-by-step arithmetic formulas.

Value of Term n=10
19
Sum of First 10 Terms100
Common Difference (d)2
First Term (a₁)1
Calculation Steps1. Formula nth Term: an = a1 + (n - 1)d, Sum: Sn = (n/2)(a1 + an) 2. Your Inputs First Term (a1) = 1, Difference (d) = 2, Target Term (n) = 10 3. Calculation Steps n-th Term: an = 1 + (10 - 1) × 2 = 19.000000 Sum: Sn = (10 / 2) × (1 + 19.000000) = 100.000000

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Master Linear Patterns

The Arithmetic Sequence calculator is a powerful tool for analyzing linear number patterns. Whether you are calculating future savings, predicting population growth, or solving algebraic progressions, this calculator provides the exact value of any term and the total sum of the sequence.

an=a1+(n−1)d,Sn=n2(a1+an)\begin{aligned} a_n = a_1 + (n-1)d, \quad S_n = \frac{n}{2}(a_1 + a_n) \end{aligned}

Where:
ana_n=
The value of the term at position n
a1a_1=
The starting value of the sequence
d=
The constant value added to each term
n=
The position of the term in the sequence
SnS_n=
The sum of the first n terms

Quick Example: What is the 10th term of 2, 5, 8...?

If you have a sequence that starts at 2 and increases by 3 each time:

  1. The First Term (a₁) is 2.
  2. The Common Difference (d) is 3.
  3. The Term Number (n) you want is 10.

Using the formula aₙ = a₁ + (n-1)d, you calculate: 2 + (10-1)×3 = 2 + 27 = 29. The 10th term is 29, and the sum of all 10 terms is 155. Our calculator handles this instantly for any numbers you provide.

Key Components of the Sequence

  1. First Term (a₁): Where the sequence begins.
  2. Common Difference (d): The amount added (or subtracted) at each step.
  3. Term Number (n): The specific position you are interested in (e.g., the 100th term).
  4. n-th Term (aₙ): The value of the number at position n.

Arithmetic Progression in Real Life

  • Financial Planning: Calculating simple interest or flat-rate savings contributions.
  • Physics: Analyzing motion with constant acceleration.
  • Computer Science: Managing loop iterations and linear data structures.
  • Music: Analyzing scales and rhythmic patterns based on equal intervals.

How to use this calculator

Enter your starting value and the difference between steps. Our tool will instantly generate the value for your target term and the cumulative sum of all terms leading up to it, complete with a full step-by-step mathematical breakdown.

Frequently Asked Questions

It is a progression of numbers where the mathematical difference between any two consecutive terms remains constant. For example: 2, 5, 8, 11 (where the constant difference is 3).

By using the formula aₙ = a₁ + (n-1)d, where a₁ represents the starting value, n represents the target position, and d represents the constant interval.

The total sum is calculated using Sₙ = (n/2) × (a₁ + aₙ), which multiplies half the number of terms by the sum of the first and last terms.

Yes. A negative interval simply creates a decreasing progression, such as 10, 7, 4, 1.

Arithmetic progressions utilize constant addition (linear growth), whereas geometric progressions utilize constant multiplication (exponential growth).

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