Arithmetic Sequence Calculator

Free Online Arithmetic Sequence Calculator

This arithmetic sequence calculator finds any missing piece of an arithmetic sequence — the nth term, the common difference, the first term, or the total number of terms — the moment you enter the values you already know. It also calculates the sum of the first n terms automatically, draws a labeled staircase diagram so you can see every term's value at a glance, and shows the complete step-by-step solution behind the answer.

Unlike a basic nth term calculator that only works in one direction, this tool lets you pick exactly what you're solving for. Know the first term, common difference, and how many terms you want? Solve for the nth term and sum. Know two terms and their positions but not the common difference? Solve for d instead. Whatever equation you're working with, the calculator rearranges the arithmetic sequence formula for you and returns an exact, fully worked answer.

What Is an Arithmetic Sequence?

An arithmetic sequence (also called an arithmetic progression) is a list of numbers in which the difference between any two consecutive terms is always the same constant value, called the common difference and written as d. For example, 2, 5, 8, 11, 14 is an arithmetic sequence because each term is exactly 3 more than the term before it — here, d = 3.

The common difference can be positive, negative, zero, or a fraction. A sequence like 20, 15, 10, 5 has a common difference of −5 because the terms are decreasing. A sequence like 4, 4, 4, 4 technically has a common difference of 0. Recognizing the common difference is the first step to solving almost any arithmetic sequence problem, and it's exactly what this calculator can find for you if you already know two terms in the sequence.

Arithmetic Sequence Formulas

Every arithmetic sequence calculation traces back to one core formula, which can be rearranged to solve for whichever variable is missing:

• nth term formula: aₙ = a₁ + (n − 1)d
• Sum of n terms: Sₙ = n/2 × (2a₁ + (n − 1)d), or equivalently Sₙ = n/2 × (a₁ + aₙ)
• Solving for the common difference: d = (aₙ − a₁) / (n − 1)
• Solving for the first term: a₁ = aₙ − (n − 1)d
• Solving for the number of terms: n = (aₙ − a₁) / d + 1

How to Use This Arithmetic Sequence Calculator

Start by choosing what you want to solve for from the dropdown at the top of the calculator: the nth term and sum, the common difference, the first term, or the number of terms. The input fields automatically relabel and show only the values you need to enter for that mode — you never have to guess which formula to rearrange by hand.

Type in your known values, and every result updates instantly: the first term, common difference, term position, the nth term, and the sum of the sequence all appear in the results panel. Below that, the staircase diagram renders the sequence as a series of bars — each one labeled with its exact value — with a small arrow between every pair of bars showing the constant step size. Finally, the step-by-step solution spells out precisely how the answer was derived from your numbers, formula by formula.

Worked Example: Finding the nth Term and Sum

Suppose a1 = 3 and the common difference d = 5, and you want to find the 10th term along with the sum of the first 10 terms. Using aₙ = a₁ + (n − 1)d, the 10th term is a₁₀ = 3 + (10 − 1) × 5 = 3 + 45 = 48.

For the sum, Sₙ = n/2 × (a₁ + aₙ) = 10/2 × (3 + 48) = 5 × 51 = 255. So the sequence runs 3, 8, 13, 18, 23, 28, 33, 38, 43, 48, and the first ten terms add up to exactly 255 — the same numbers this calculator's default example uses.

Worked Example: Finding the Common Difference

Now suppose you know a1 = 4 and the 6th term a6 = 24, but you don't know the common difference. Rearranging the formula gives d = (aₙ − a₁) / (n − 1) = (24 − 4) / (6 − 1) = 20 / 5 = 4.

That means the sequence is 4, 8, 12, 16, 20, 24 — each term increases by exactly 4. Switching the calculator's 'Solve for' dropdown to Common Difference and entering a1 = 4, a6 = 24, and n = 6 reproduces this exact result, along with the full working.

Worked Example: Finding the Number of Terms

Sometimes you know the first term and the common difference, along with a target value, but you need to know which position that value occupies in the sequence. Suppose a1 = 7, the common difference d = 4, and you want to know which term equals 79. Rearranging the formula gives n = (aₙ − a₁) / d + 1 = (79 − 7) / 4 + 1 = 72 / 4 + 1 = 18 + 1 = 19.

So 79 is the 19th term of the sequence 7, 11, 15, 19, ... and so on. If this calculation had produced a decimal instead of a whole number, that would mean 79 simply isn't a term of this particular sequence at all — the calculator automatically flags that case for you instead of returning a misleading fractional position.

Worked Example: Finding the First Term

Finally, suppose you're told the common difference is d = −3, the 8th term is a8 = −11, and you need to find where the sequence started. Rearranging gives a1 = aₙ − (n − 1)d = −11 − (8 − 1) × (−3) = −11 − (−21) = −11 + 21 = 10.

So the sequence begins at a1 = 10 and decreases by 3 each time: 10, 7, 4, 1, −2, −5, −8, −11 — confirming that −11 is indeed the 8th term. This mode is especially useful in word problems that describe a sequence's ending value or a later term but never state where it actually began.

Understanding the Staircase Diagram

The diagram is designed so you can understand the sequence just by looking at it, without needing to read a single number in text. Each bar represents one term of the sequence, colored by role — amber for the first term (a₁), indigo for the middle terms, and green for the target nth term (aₙ) — and every bar is labeled directly with its own value, so nothing is hidden behind an abstract axis or scale.

The pink arrows between consecutive bars show the common difference, d, repeated at every single step — visually proving that the sequence climbs (or falls) by the exact same amount every time. For sequences with many terms, the diagram automatically compresses the middle into an ellipsis so the first few terms and the final target term (aₙ) stay clearly visible without the chart becoming cluttered.

Arithmetic vs. Geometric Sequences

It's easy to confuse arithmetic sequences with geometric sequences, but the difference is simple: in an arithmetic sequence, consecutive terms differ by a constant amount (addition), while in a geometric sequence, consecutive terms differ by a constant ratio (multiplication). For example, 2, 4, 6, 8 is arithmetic (add 2 each time), while 2, 4, 8, 16 is geometric (multiply by 2 each time).

If your sequence is growing or shrinking by multiplying rather than adding, use our dedicated Geometric Sequence Calculator instead, which applies the aₙ = a₁ × r^(n−1) formula rather than the addition-based formula used here.

Real-World Applications of Arithmetic Sequences

Arithmetic sequences show up constantly outside the math classroom, anywhere something increases or decreases by a fixed, predictable amount:

• Savings plans where a fixed amount is deposited every period, such as saving $50 more each month than the last.
• Seating arrangements in a theater or stadium, where each row has a fixed number of additional seats compared to the row before it.
• Stacking objects like pipes, logs, or crates in a pyramid, where each layer has a constant number fewer (or more) than the layer below.
• Salary increments that add a fixed raise each year, letting you project future salary using the nth term formula.
• Simple interest calculations, where the balance increases by the exact same dollar amount every compounding period.
• Physics problems involving constant acceleration, such as the distance covered in each successive second of uniformly accelerated motion.

Tips for Working with Arithmetic Sequences

Always double-check the sign of your common difference — a decreasing sequence has a negative d, and mixing up the sign is the most common source of errors when solving by hand. Remember that n always represents a whole-number position (1st term, 2nd term, and so on), so if a calculated value of n comes out as a decimal, it usually means the target value you entered isn't actually part of that sequence.

When solving for the sum, it's often faster and less error-prone to use Sₙ = n/2 × (a₁ + aₙ) once you already know the nth term, rather than expanding the longer Sₙ = n/2 × (2a₁ + (n − 1)d) version — both formulas always give the identical result, so use whichever one fits the values you already have on hand.