Math Tools
Arithmetic Sequence Calculator
Find the nth term, generate arithmetic sequence terms, and calculate arithmetic series sums using the arithmetic sequence formulas.
Arithmetic Sequence Calculator
Find the nth term, generate terms, and calculate the sum of an arithmetic sequence using the first term, common difference, and term number.
Formulas
aₙ = a + (n - 1)d
Sₙ = n/2 × (2a + (n - 1)d)
Results
Nth term
29
Sum of first n terms
155
Sequence
2, 5, 8, 11, 14, 17, 20, 23, 26, 29
First 10 terms table
| n | Term value |
|---|---|
| 1 | 2 |
| 2 | 5 |
| 3 | 8 |
| 4 | 11 |
| 5 | 14 |
| 6 | 17 |
| 7 | 20 |
| 8 | 23 |
| 9 | 26 |
| 10 | 29 |
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What this calculator does
An arithmetic sequence changes by the same amount each time. This calculator finds the nth term, generates the sequence, and calculates the sum of the first n terms.
What is an arithmetic sequence?
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is always the same. That constant change is called the common difference.
For example, in the sequence 2, 5, 8, 11, 14, each term increases by 3. Because the change stays constant, this is an arithmetic sequence.
Arithmetic sequences are common in algebra, school math, exam practice, finance, and pattern recognition problems. This arithmetic sequence calculator helps you find terms quickly without doing each step by hand.
Arithmetic sequence formulas
Nth term formula
The arithmetic sequence nth term formula is:
aₙ = a + (n - 1)d
Here, a is the first term, d is the common difference, and n is the term number.
Sum of arithmetic sequence formula
The sum of the first n terms is:
Sₙ = n/2 × (2a + (n - 1)d)
This formula is useful when you need the arithmetic series sum instead of listing every term manually.
Arithmetic sequence formula table
| Formula | Expression | Use |
|---|---|---|
| Nth term formula | aₙ = a + (n - 1)d | Find a specific term in an arithmetic sequence. |
| Sum formula | Sₙ = n/2 × (2a + (n - 1)d) | Find the sum of the first n terms. |
| Common difference | d = a₂ - a₁ | Check whether a sequence increases or decreases by a fixed amount. |
Arithmetic sequence examples table
These examples show how the arithmetic sequence nth term and arithmetic series sum are calculated for different values.
| First term (a) | Common difference (d) | n | Nth term | Sum of first n terms |
|---|---|---|---|---|
| 2 | 3 | 10 | 29 | 155 |
| 5 | 5 | 8 | 40 | 180 |
| 20 | -2 | 6 | 10 | 90 |
| -4 | 6 | 7 | 32 | 98 |
How to use this arithmetic sequence calculator
Enter the first term, the common difference, and the number of terms. The calculator will instantly find the nth term, the sum of the first n terms, and generate the sequence values for you.
This makes it useful for homework, exam revision, algebra practice, and quick verification of arithmetic progression problems.
Arithmetic sequence worked example
Suppose the first term is 2, the common difference is 3, and you want the 10th term.
a₁₀ = 2 + (10 - 1) × 3 = 2 + 27 = 29
The first 10 terms are:2, 5, 8, 11, 14, 17, 20, 23, 26, 29
The sum of the first 10 terms is:155
Arithmetic sequence vs arithmetic series
An arithmetic sequence is the list of terms, such as 2, 5, 8, 11, and 14. An arithmetic series is the sum of those terms.
This means the sequence describes the pattern, while the series describes the total when the terms are added together. This calculator gives you both the generated sequence and the arithmetic series sum.
Where arithmetic sequences are used
Arithmetic sequences appear in school math, algebra, accounting, budgeting, installment plans, schedules, and any repeated pattern where values rise or fall by the same fixed amount.
They are also useful in understanding arithmetic series, linear growth, and step-based progressions in both academic and practical settings.
Arithmetic sequence calculator FAQ and common questions
How do I find the nth term of an arithmetic sequence?
Use the formula aₙ = a + (n - 1)d, where a is the first term and d is the common difference.
How do I find the sum of an arithmetic sequence?
Use the formula Sₙ = n/2 × (2a + (n - 1)d) to find the sum of the first n terms.
What is the common difference?
The common difference is the fixed amount added to or subtracted from one term to get the next term.
Can the common difference be negative?
Yes. A negative common difference means the arithmetic sequence decreases by the same amount each step.
What is the difference between an arithmetic sequence and an arithmetic series?
An arithmetic sequence is the list of terms, while an arithmetic series is the sum of those terms.
How do I know if a sequence is arithmetic?
A sequence is arithmetic if the difference between every pair of consecutive terms is the same throughout the pattern.
Can this arithmetic sequence calculator generate terms automatically?
Yes. After entering the first term, common difference, and number of terms, the calculator generates the arithmetic sequence values automatically.
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