Arithmetic Sequence Calculator
Calculate the nth term, sum, and common difference of an arithmetic sequence. An arithmetic sequence is a sequence where each term differs from the previous by a constant value called the common difference.
This free online arithmetic sequence calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.
The starting value of the sequence
The constant value added to each term
Which term to calculate (n ≥ 1)
Results
nth Term (aₙ)
19
Sum of First n Terms (Sₙ)
100
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Arithmetic Sequence Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.
Review your inputs
Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Arithmetic Sequence Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
Formula Reference
Arithmetic Sequence Calculator Formula
See calculator inputs for the governing equation
Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.
When to Use This Calculator
- •Use the Arithmetic Sequence Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
- •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
- •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
- •Use it to explore the behavior of mathematical functions across a range of inputs.
About This Calculator
The Arithmetic Sequence Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Calculate the nth term, sum, and common difference of an arithmetic sequence. An arithmetic sequence is a sequence where each term differs from the previous by a constant value called the common difference. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.
About Arithmetic Sequence Calculator
An arithmetic sequence (also called an arithmetic progression) is one of the most fundamental concepts in mathematics. It is a sequence of numbers in which the difference between consecutive terms remains constant. This constant difference, denoted d, determines whether the sequence increases, decreases, or stays flat. Arithmetic sequences appear everywhere: house numbers on a street, seating rows in a theater, salary increments, and even in the spacing of musical notes on a linear scale. Understanding arithmetic sequences is essential for algebra, calculus, and many real-world applications including financial planning, scheduling, and engineering design. The two key quantities are the nth term formula, which lets you jump directly to any term without listing all preceding ones, and the partial sum formula, which efficiently totals any number of consecutive terms. Carl Friedrich Gauss famously exploited the sum formula as a schoolboy when asked to add the integers from 1 to 100, pairing the first and last terms to get 101, then multiplying by 50 pairs to arrive at 5050. This calculator automates both computations so you can explore arithmetic sequences of any size instantly.
The Math Behind It
Formula Reference
nth Term Formula
aₙ = a₁ + (n − 1)d
Variables: a₁ = first term, d = common difference, n = term number
Sum Formula
Sₙ = n/2 × (2a₁ + (n − 1)d)
Variables: n = number of terms, a₁ = first term, d = common difference
Worked Examples
Example 1: Find the 20th Term
Given a₁ = 3 and d = 5, find a₂₀.
a₂₀ = 98
Example 2: Sum of First 50 Terms
Given a₁ = 2 and d = 3, find S₅₀.
S₅₀ = 3775
Common Mistakes & Tips
- !Using n instead of (n − 1) in the nth term formula — remember the first term already occupies position 1.
- !Forgetting that d can be negative for decreasing sequences.
- !Confusing the sum formula Sₙ with the nth term formula aₙ.
- !Using n = 0 when the sequence is 1-indexed.
Related Concepts
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Frequently Asked Questions
What is the difference between an arithmetic sequence and an arithmetic series?
An arithmetic sequence is the ordered list of terms (e.g., 2, 5, 8, 11, …), while an arithmetic series is the sum of those terms (e.g., 2 + 5 + 8 + 11 + …). The sequence focuses on individual terms; the series focuses on their cumulative total.
Can the common difference be zero?
Yes. When d = 0 every term equals the first term, producing a constant sequence such as 7, 7, 7, 7, ….
How do I find the common difference if I know two terms?
Use d = (aₘ − aₖ) / (m − k), where aₘ and aₖ are two known terms at positions m and k.