Arithmetic Sequence Calculator
Find the nth Term, Sum ($S_n$), and Difference
$$ a_n = a_1 + (n-1)d $$
First Term ($a_1$)
Difference ($d$)
Term Index ($n$)
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nth Term ($a_n$)
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Sum ($S_n$)
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Sequence Graph
Detailed Steps
First $n$ Terms
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By Prof. David Anderson
Professor of Applied Mathematics | 20+ Years Teaching Experience
“If Mathematics is a language, then Arithmetic Sequences are its simplest sentence structure. They represent perfect linear growth—like climbing a ladder one rung at a time. Many students memorize the formulas but forget the logic. I built this Arithmetic Sequence Calculator to not only give you the answers but to visualize the ‘steps’ of the ladder so you truly understand the pattern.”
The Professor’s Guide to Using an Arithmetic Sequence Calculator: Formulas, Series, and Sums
A Complete Handbook on Nth Terms, Partial Sums, and Real-World Applications
⚡ Key Takeaways for Students
- An Arithmetic Sequence (or Arithmetic Progression) adds a constant Common Difference ($d$) at each step.
- Nth Term Formula: $a_n = a_1 + (n-1)d$. Use this to find any specific number in the list.
- Arithmetic Series Formula: $S_n = \frac{n}{2}(a_1 + a_n)$. Use this to find the total sum.
- Graphing: Arithmetic sequences always form a linear (straight line) graph.
Welcome to the definitive guide on Arithmetic Progressions (A.P.). Whether you are solving for a homework assignment or calculating simple interest, understanding linear sequences is fundamental. Unlike geometric sequences which explode exponentially, arithmetic sequences grow steadily and predictably.
Our Arithmetic Sequence Calculator above is designed to handle the two most common tasks: finding the value of a specific term ($a_n$) and calculating the Sum of Arithmetic Sequence ($S_n$).
1. Anatomy of an Arithmetic Sequence
To use the calculator effectively, you must understand the three key variables that define every linear sequence.
| Symbol | Name | Definition | Example (5, 8, 11…) |
|---|---|---|---|
| $a_1$ | First Term | The starting value of the sequence. | $5$ |
| $d$ | Common Difference | The amount added to get the next term ($a_2 – a_1$). | $8 – 5 = 3$ |
| $n$ | Term Position | The index or “rank” of the term you want. | $n=10$ |
2. Calculating the Nth Term of an Arithmetic Sequence
How do we find the 100th term without adding 3 ninety-nine times? We use the Explicit Formula.
Explicit Formula
$$ a_n = a_1 + (n-1)d $$
Why $n-1$? Think of a fence. If you have 10 fence posts ($n=10$), there are only 9 spaces ($n-1$) between them. The “spaces” represent the common difference $d$. To get to the 10th term, you add the difference 9 times. This logic is built into our Nth Term Calculator.
3. The Sum of Arithmetic Series (The Gauss Method)
This is my favorite mathematical story. When the famous mathematician Carl Friedrich Gauss was a child, his teacher asked the class to add the numbers from 1 to 100. Gauss solved it in seconds.
He noticed that if you pair the first and last numbers, the sum is constant:
$1 + 100 = 101$
$2 + 99 = 101$
$3 + 98 = 101$
…
Since there are 100 numbers, there are 50 pairs. So, $50 \times 101 = 5050$. This logic gives us the Arithmetic Series Formula used by our calculator:
$$ S_n = \frac{n}{2}(a_1 + a_n) $$
Or, if you don’t know the last term ($a_n$), substitute the explicit formula in:
$$ S_n = \frac{n}{2}[2a_1 + (n-1)d] $$
4. Recursive vs. Explicit Formulas
Math textbooks often ask for the “Recursive Formula.” This is simply a rule that tells you how to get the next term from the current term.
- Recursive Formula: $a_n = a_{n-1} + d$ (Great for computers, bad for humans finding the 100th term).
- Explicit Formula: $a_n = a_1 + (n-1)d$ (Great for humans jumping straight to the answer).
5. Real-World Applications
Example: Stadium Seating
A stadium section has 20 seats in the first row ($a_1=20$). Each subsequent row has 2 more seats than the previous one ($d=2$). How many seats are in the 50th row ($a_{50}$)? What is the total capacity of the 50 rows ($S_{50}$)?
- Find Row 50: $a_{50} = 20 + (49)(2) = 20 + 98 = 118$ seats.
- Total Capacity: $S_{50} = \frac{50}{2}(20 + 118) = 25(138) = 3,450$ seats.
6. Frequently Asked Questions (FAQ)
Can the common difference be negative?
Yes. If $d$ is negative (e.g., $d = -5$), the sequence is Decreasing (e.g., 100, 95, 90…). Our calculator handles negative differences perfectly.
What is the difference between Sequence and Series?
A Sequence is the list of numbers itself (comma-separated). A Series is the sum of those numbers (plus signs).
Sequence: 2, 4, 6
Series: 2 + 4 + 6 = 12
Sequence: 2, 4, 6
Series: 2 + 4 + 6 = 12
Is Simple Interest an arithmetic sequence?
Yes! Simple interest adds a fixed amount of money (based on the principal) every year. This is a classic linear growth pattern modeled by arithmetic sequences. (Compound interest, however, is geometric).
References & Further Reading
- Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage Learning. (Chapter 11: Sequences).
- OpenStax. “Arithmetic Sequences.” Read Text
- Khan Academy. “Arithmetic sequences and series.” Watch Video
Calculate Your Linear Growth Now
Stop adding numbers manually. Use our free Arithmetic Sequence Calculator to instantly find the Nth term, Partial Sums, and visualize the progression graph.
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