About this calculator
Arithmetic sequence is one of the most basic and important types of sequence in mathematics. In an arithmetic sequence, starting from the second term, the difference between each term and the previous term is equal to the same constant. This constant is called the tolerance (d). The general formula of the arithmetic sequence is aₙ = a₁ + (n-1)d, and the sum formula of the first n terms is Sₙ = n(a₁+aₙ)/2 or Sₙ = na₁ + n(n-1)d/2.
Arithmetic sequence can be seen everywhere in daily life. The calculation of simple interest on bank deposits, fixed growth of wages, equidistantly arranged seat numbers, prices increasing in fixed steps, etc. are all practical applications of arithmetic sequence. In the fields of mathematics, physics, economics and other fields, arithmetic sequence is an important tool for solving problems.
Our arithmetic sequence calculator can help you quickly calculate any term of the arithmetic sequence, the sum of the first n terms, tolerance and other parameters. Whether it is students doing math homework, teachers setting questions, or data analysis in actual work, this calculator can provide accurate and fast calculation results. Supports positive numbers, negative numbers, decimals and fractions to meet various calculation needs.
What it calculates
The arithmetic sequence calculator finds the nth term, common difference, first term, number of terms, and sum of an arithmetic sequence.
Formula
- nth term: a_n = a_1 + (n - 1)d.
- Sum: S_n = n(a_1 + a_n) / 2.
- Also S_n = n(2a_1 + (n - 1)d) / 2.
Inputs
- First term a_1.
- Common difference d.
- Term count n or target term.
Example
| Input | Result | Note |
|---|---|---|
| a1=2,d=3,n=5 | a5=14 | 2,5,8,11,14 |
| a1=4,d=2,n=10 | S10=130 | Sum of first 10 terms |
| a1=10,d=-1,n=4 | a4=7 | Decreasing sequence |
How to interpret the result
An arithmetic sequence has the same difference between neighboring terms. Positive d increases, negative d decreases, and d=0 gives constant terms.
Common mistakes
- n usually starts at 1.
- Do not confuse common difference with common ratio.
- For sums, confirm you are summing the first n terms.
How to use
Using the Arithmetic Sequence Calculator is easy and intuitive. First, identify the parameters you already know. Usually you need to know at least three parameters of the leading term (a₁), tolerance (d) and number of terms (n) to calculate other unknown quantities.
**Basic calculation steps:** 1. Enter the first term a₁ (the first number in the sequence) 2. Enter the tolerance d (the difference between two adjacent items) 3. Enter the number of items n (to calculate the number of items or the sum of the previous items) 4. Select the calculation type: general term (value of the nth term) or sum (sum of the first n terms) 5. Click the "Calculate" button to get the result
**Example 1:** It is known that the first term a₁=3 and the tolerance d=2, find the 10th term. After inputting, it is calculated: a₁₀ = 3 + (10-1)×2 = 21.
**Example 2:** It is known that the first term a₁=5 and the tolerance d=3, find the sum of the first 20 terms. Calculated: S₂₀ = 20×5 + 20×19×3/2 = 670.
The calculator also supports reverse calculations. If you know the value, leading term, and number of terms of an item, you can work backwards to deduce the tolerance. This flexibility allows you to solve a variety of arithmetic sequence problems.
Main features
• General term calculation: quickly calculate the value of the nth term based on the first term, tolerance and number of terms • Sum calculation: Calculate the sum of the first n terms of the arithmetic sequence • Reverse solution: known partial parameters, inverse unknown parameters (such as tolerance, first term) • Formula display: displays detailed calculation formulas and derivation processes • Step description: Show the calculation process of each step to facilitate learning and understanding. • Multiple inputs: supports integers, decimals, negative numbers and fractions • Sequence display: List the first several items of the sequence to visually display the rules • Graphical display: draw images of the sequence and visualize the changing trend of the sequence • Parameter verification: automatically checks the plausibility of input parameters • Totally free: no registration required, unlimited use
Use cases
• Mathematics learning: students practice the concept of arithmetic sequences and verify homework answers • Exam preparation: quickly check calculation results and improve problem-solving efficiency • Teaching assistance: teachers set questions, correct homework, and explain example questions • Salary calculation: Calculate total salary in fixed amount increments • Deposit interest: Calculate the sum of principal and interest on simple interest deposits • Seat Number: Calculates equidistant seat numbers • Price analysis: Analyze price series that change in fixed steps • Engineering surveying: Calculation of values at equally spaced measurement points • Data analysis: Analyze linear growth data trends • Competition training: Solving sequence problems in mathematics competitions