Arithmetic Sequence Calculator: Find Any Term Step by Step

What Is an Arithmetic Sequence?

An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is always the same. That constant difference is called the common difference, written as d. For example, 3, 7, 11, 15, 19 is an arithmetic sequence because each term is 4 more than the one before it: 7−3=4, 11−7=4, 15−11=4, and 19−15=4. Compare that to 2, 4, 8, 16, where each term is multiplied (not added) by 2—that's a geometric sequence, not an arithmetic one. Recognizing which type of pattern you're looking at is the first step before you plug anything into a formula or calculator.

If subtracting consecutive terms doesn't give the same answer every time, you're not dealing with a true arithmetic sequence.

The Arithmetic Sequence Formula Explained

Every arithmetic sequence can be described with one formula: aₙ = a₁ + (n − 1)d, where a₁ is the first term, d is the common difference, and n is the position of the term you want (1st, 2nd, 15th, and so on). This formula works because to get from the first term to the nth term, you add the common difference exactly (n − 1) times—not n times, since the first term itself doesn't need any adjustment. Mixing up n and (n − 1) is the single most common error students make, so it's worth memorizing the formula with the subtraction built in.

How Does an Arithmetic Sequence Calculator Work?

Behind the scenes, an arithmetic sequence calculator is just running the same formula you'd use by hand: aₙ = a₁ + (n − 1)d for individual terms, and Sₙ = n/2 × (2a₁ + (n − 1)d) for the sum of the first n terms. You type in the first term, the common difference, and how many terms you want, and the calculator substitutes those values automatically, handles the order of operations correctly, and returns the result instantly. The real value of a calculator isn't replacing the math—it's letting you check your own hand-worked answer in seconds and catching arithmetic slips before they cost you points on a test.

A calculator gives you the answer fast; working one example by hand tells you whether you actually understand why.

Finding the Common Difference

The common difference is the backbone of every arithmetic sequence calculation, so it's worth practicing until it's automatic. Take the sequence 5, 12, 19, 26, 33. Subtracting consecutive terms gives 12−5=7, 19−12=7, 26−19=7, and 33−26=7, so d=7. If a problem only gives you two non-consecutive terms, such as the 3rd term (19) and the 6th term (40), you can still find d by dividing the difference in value by the difference in position: (40−19)÷(6−3) = 21÷3 = 7, which matches.

How to Find the nth Term: Step-by-Step Example

Suppose a sequence starts at a₁=4 with a common difference of d=6, and you need the 15th term. Using aₙ = a₁ + (n − 1)d: a₁₅ = 4 + (15 − 1) × 6 = 4 + 14 × 6 = 4 + 84 = 88. To check the answer, list the first few terms and confirm the pattern holds: 4, 10, 16, 22, 28... each jumping by 6, which lines up with the formula's result of 88 at position 15. This two-step habit—compute, then verify with a partial list—catches the majority of sign and arithmetic errors before they become wrong answers.

Calculating the Sum of an Arithmetic Sequence

To add up the first n terms of an arithmetic sequence without listing every number, use Sₙ = n/2 × (a₁ + aₙ), or equivalently Sₙ = n/2 × (2a₁ + (n − 1)d) when you don't already know the last term. Take the sequence 3, 7, 11, 15... and find the sum of the first 20 terms. First find a₂₀ = 3 + (20 − 1) × 4 = 3 + 76 = 79. Then S₂₀ = 20/2 × (3 + 79) = 10 × 82 = 820. This formula works because pairing the first and last terms, the second and second-to-last, and so on, always produces the same sum—a trick famously attributed to a young Carl Friedrich Gauss.

Pairing the first and last term is the same trick Gauss reportedly used as a schoolboy to sum 1 through 100 in seconds.

Worked Example: Solving a Real Word Problem

Suppose you save $50 in the first week, then increase your savings by $10 every week after that, for 12 weeks total. This is an arithmetic sequence with a₁=50 and d=10. The amount saved in week 12 is a₁₂ = 50 + (12 − 1) × 10 = 50 + 110 = 160. To find the total saved across all 12 weeks, use the sum formula: S₁₂ = 12/2 × (50 + 160) = 6 × 210 = 1,260. As a sanity check, the average weekly savings should sit halfway between the first ($50) and last ($160) amounts, which is (50+160)/2 = $105—multiplying that average by 12 weeks gives 105 × 12 = 1,260, matching the formula's result exactly.

What's the Difference Between Arithmetic and Geometric Sequences?

Arithmetic sequences grow by addition—each term is the previous term plus a fixed common difference, like 5, 9, 13, 17 where d=4. Geometric sequences grow by multiplication instead—each term is the previous term times a fixed common ratio, like 2, 4, 8, 16 where the ratio r=2. The formulas look similar but behave very differently: an arithmetic sequence's terms increase (or decrease) in a straight line, while a geometric sequence's terms curve upward or downward exponentially. Before reaching for any calculator, check whether subtraction or division produces the constant pattern—that tells you which formula family applies.

Common Mistakes When Using an Arithmetic Sequence Calculator

Even with a reliable arithmetic sequence calculator, a few input errors show up constantly. The most frequent is using n instead of (n − 1) in the exponent-style position, which shifts every answer by exactly one common difference. Another is entering the wrong sign for d—subtracting in the wrong order turns a decreasing sequence into an increasing one, or vice versa. Students also sometimes confuse the term value with the term position, plugging a₃=19 into the slot meant for n. Finally, when working with the sum formula, forgetting to find aₙ first (when it isn't already given) leads to an incomplete calculation.

Can Arithmetic Sequences Have Negative or Fractional Differences?

Yes—the common difference can be negative, a fraction, or a decimal, and the formula works exactly the same way. Consider the decreasing sequence 10, 7, 4, 1, −2, where each term drops by 3, so d=−3. Using the formula to find the 8th term: a₈ = 10 + (8 − 1) × (−3) = 10 + (−21) = −11. Fractional differences work identically: in the sequence 1, 1.5, 2, 2.5, the common difference is d=0.5, and the 10th term is a₁₀ = 1 + (10 − 1) × 0.5 = 1 + 4.5 = 5.5. The formula never changes—only the sign and size of d determine whether the sequence rises, falls, or moves in small fractional steps.

Practice Problems to Test Your Understanding

Try these three problems by hand, then verify with a calculator. Problem 1: If a₁=2 and d=5, find a₁₀. Answer: a₁₀ = 2 + 9 × 5 = 2 + 45 = 47. Problem 2: For the sequence 9, 4, −1, −6..., find the 8th term. First find d = 4 − 9 = −5, then a₈ = 9 + 7 × (−5) = 9 − 35 = −26. Problem 3: Find the sum of the first 15 terms of the sequence 6, 10, 14, 18... First find a₁₅ = 6 + 14 × 4 = 6 + 56 = 62, then S₁₅ = 15/2 × (6 + 62) = 7.5 × 68 = 510. Work through each one step by step before checking the final number—the process matters more than the answer alone.

Using Solvify's Arithmetic Sequence Calculator to Check Your Work

Once you've worked through the formulas by hand, Solvify's AI Solve tool can confirm your answer and show its own step-by-step breakdown, which is especially useful when a homework set has a dozen similar problems to grind through. Snap a photo of a printed sequence problem with Smart Scan, and Solvify identifies the first term and common difference automatically, then walks through the nth-term or sum calculation the same way this article did. If your answer doesn't match, the follow-up question feature lets you ask exactly where the two approaches diverged instead of just seeing a different final number.