What this calculator does
Find any number in a list that goes up (or down) by the same step each time, and the total so far. That kind of list is an arithmetic sequence. For example, 3, 7, 11, 15, … adds 4 each time. Each number in the list is a term, and the fixed step is the common difference, d.
Seat rows that gain two seats each are an arithmetic sequence. So is a savings jar topped up by the same amount every week, and so are the odd numbers. Give the first term and the common difference, and this calculator finds any term. It also adds up all the terms to that point. Or it can work backward and tell you which position a value holds. A table lists the first terms with a running total, so you can see the pattern for yourself.
Where it's used
Why this math matters outside the classroom:
- Stadium seating. A section whose first row has 20 seats and each row adds 2 more has 20 + 2 × 24 = 68 seats in row 25.
- Savings plans. Saving $10 in week 1 and $5 more each week after puts away 52 × (10 + 265) ÷ 2 = $7,150 in a year.
- Stacking cans. A display with 1 can on top and one more in each row below holds 10 × 11 ÷ 2 = 55 cans in 10 rows.
- Taxi fares. A fare of $3.00 at the start that rises $0.50 every quarter mile reaches 3 + 0.5 × 12 = $9.00 after 3 miles.
- Training plans. A runner who starts at 2 miles and adds half a mile each week is running 2 + 0.5 × 9 = 6.5 miles in week 10.
Grade level and standards: Grades 9–11; Common Core State Standards for Mathematics HSF.BF.A.2, HSF.IF.A.3. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Choose whether to find the nth term and sum, or the position of a given value.
- Enter the first term a₁ and the common difference d. Either can be negative or a decimal.
- Enter the term number n, or the term value whose position you want.
- Read the result. Then check the steps and the table of the first terms.
What the inputs mean
- First term (a₁)
- The first term of the sequence, at position 1.
- Common difference (d)
- The difference between any term and the one before it. To find it, subtract the first term from the second. A negative d makes the sequence go down.
- Term number (n)
- The position you want, counting the first term as 1. Whole numbers up to one billion.
- Term value (aₙ)
- A value to look for. The calculator tells you its position if it's in the sequence, or explains why it isn't.
Formula and method
aₙ = a₁ + (n − 1) × d Sₙ = n/2 × (a₁ + aₙ) Sₙ = n/2 × (2a₁ + (n − 1) × d) n = (aₙ − a₁) / d + 1
a₁ is the first term, d is the common difference, and n is a position in the list. aₙ is the term at position n, and Sₙ is the sum of the first n terms.
To reach position n, you start at a₁ and take n − 1 steps of size d. That gives the nth-term formula.
For the sum, pair the first term with the last, the second with the second-to-last, and so on. Every pair adds up to the same total, a₁ + aₙ, and there are n/2 pairs. This is the trick often credited to a young Gauss for adding 1 to 100.
Solving the nth-term formula for n tells you where a value sits. If the answer isn't a whole number of at least 1, the value isn't in the sequence.
Worked example
A theater's first row has 3 seats and each row behind it has 4 more. How many seats are in row 10, and how many seats are in the first 10 rows?
- The nth term: a₁₀ = 3 + (10 − 1) × 4 = 3 + 36 = 39 seats.
- The sum: S₁₀ = 10/2 × (3 + 39) = 5 × 42 = 210 seats.
- Working backward, a row with 39 seats is at position n = (39 − 3) / 4 + 1 = 10.
Result: Row 10 has 39 seats, and rows 1 to 10 hold 210 seats in total.
Understanding the result
The terms grow in a straight line. If you plot each term against its position, the points sit on a line whose slope (steepness) is d.
When d isn't zero, the sum grows roughly with the square of n. So totals climb much faster than the terms themselves.
If the position comes out as a fraction, the value falls between two terms. It isn't part of the sequence.
Assumptions and limitations
- Positions start at 1 for the first term.
- The common difference is the same between every pair of neighboring terms.
- Computers store decimals with tiny rounding. So very long sequences with decimal steps can show small errors in the last digits.
- The table shows at most the first 20 terms.
- With a common difference of 0, every term is the same, so a value has no single position. The calculator explains this instead of giving a number.
Frequently asked questions
How do I find the common difference?
Subtract any term from the one that follows it. For 12, 9, 6, 3, …, d = 9 − 12 = −3. Check a second pair to make sure the sequence really is arithmetic.
What is the difference between an arithmetic sequence and an arithmetic series?
The sequence is the list of terms. The series is what you get when you add them up. Sₙ, the sum of the first n terms, is called a partial sum of the series.
Can the sum be smaller than the first term?
Yes. When d is negative, the later terms can drop below zero and pull the total down. For 20, 17, 14, …, the first 8 terms end at −1 and add up to 76.
How is this different from a geometric sequence?
An arithmetic sequence adds the same amount each step. A geometric sequence multiplies by the same number each step. So arithmetic sequences grow in a straight line (linearly), while geometric ones change by the same factor each step (exponentially).
How this calculator is checked
Method: Standard closed forms for the nth term and partial sum of an arithmetic progression. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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