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Arithmetic Sequence Calculator

An arithmetic sequence is a list of numbers where each term increases or decreases by a fixed amount called the common difference. This calculator finds any term in the sequence, computes partial sums, and identifies missing values when you provide partial information. Engineers, statisticians, and students use it to model evenly-spaced progressions in physics, finance, and data analysis.

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Creators Mateusz Mucha, BEng
3,447 people find this calculator helpful

What Makes an Arithmetic Sequence

An arithmetic sequence is fundamentally a mathematical pattern where consecutive terms maintain a constant gap between them. This gap—known as the common difference—remains identical no matter where you are in the sequence. For instance, the sequence 2, 5, 8, 11, 14 has a common difference of 3 because each term exceeds the previous one by exactly that amount.

Sequences can be finite, containing a specific number of terms, or infinite, continuing indefinitely. The common difference itself can be positive (causing the sequence to increase), negative (causing it to decrease), or even fractional. A sequence like 10, 9.5, 9, 8.5, 8 demonstrates a negative difference of −0.5, while 7, 14, 21, 28 shows a larger positive difference of 7.

Distinguishing between a sequence and a series is essential: a sequence is simply the ordered list of numbers, whereas a series refers to their sum. When you add up terms from an arithmetic sequence, you create an arithmetic series.

Core Formula for Finding Terms

Rather than writing out dozens of terms manually, you can directly calculate any position using the arithmetic sequence formula. Knowing the first term and common difference allows you to leap straight to the 50th, 100th, or any other term without computing all intermediate values.

aₙ = a₁ + (n − 1) × d

aₙ = aₘ + (n − m) × d

  • aₙ — The value of the term at position n
  • a₁ — The first term of the sequence
  • d — The common difference between consecutive terms
  • n — The position of the term you want to find
  • aₘ — The value of any known term at position m

Summing an Arithmetic Series

Once you understand individual terms, calculating their sum becomes powerful. Rather than adding 100 numbers one by one, there's an elegant shortcut: pair the smallest and largest terms, then multiply by how many pairs exist.

If you sum the first and last term, you always get the same result as the second and second-to-last. For example, in 3, 5, 7, 9, 11: pairing 3 + 11 = 14 and 5 + 9 = 14 reveals the pattern.

The formula becomes:

S = n/2 × (a₁ + aₙ) = n/2 × (2a₁ + (n − 1) × d)

where S is the sum, n is the number of terms, a₁ is the first term, aₙ is the last term, and d is the common difference. An infinite arithmetic sequence with a non-zero difference always sums to infinity, regardless of sign.

Arithmetic vs. Other Sequences

Arithmetic sequences differ fundamentally from geometric sequences, which use multiplication instead of addition. In a geometric sequence like 2, 4, 8, 16, each term multiplies the previous by a constant ratio of 2. The key test: if the difference between consecutive terms changes, you don't have an arithmetic sequence.

A special hybrid exists called the arithmetico-geometric sequence, created by multiplying corresponding terms from an arithmetic and geometric progression. While your calculator focuses on pure arithmetic patterns, recognizing these distinctions helps you classify any sequence correctly.

Common Pitfalls and Practical Tips

Avoid these mistakes when working with arithmetic sequences.

  1. Confusing Sequence and Series — A sequence is the list itself (2, 4, 6, 8). A series is the sum (2 + 4 + 6 + 8 = 20). Many people use these interchangeably, but calculators often require you to specify which you need. Know which output you're after before entering data.
  2. Forgetting That d Can Be Negative — A decreasing sequence like 20, 17, 14, 11 has d = −3, not +3. Miscalculating the sign leads to wildly incorrect predictions. Always subtract: later term minus earlier term to capture the true direction.
  3. Index Numbering Confusion — Some sequences start at index 0, others at index 1. This calculator uses index 1 as the first term (a₁). Ensure you're counting positions consistently; the 5th term is a₅, not a₄. Off-by-one errors propagate through all subsequent calculations.
  4. Testing with Adjacent Pairs Only — When identifying the common difference, verify it's truly constant by checking multiple non-adjacent pairs, not just the first two terms. A sequence might match for the first few terms but diverge later if it's not genuinely arithmetic.

Frequently Asked Questions

How do I locate a specific term in an arithmetic sequence?

Multiply the common difference d by (n − 1), then add the result to the first term a₁. The formula aₙ = a₁ + (n − 1) × d gives you any term's value directly. For example, if a₁ = 3 and d = 2, the 10th term is 3 + (10 − 1) × 2 = 21. This avoids tedious counting and saves time for large position values.

What is the common difference, and how do I find it?

The common difference is the fixed amount by which each term differs from the previous one. To find it, subtract any earlier term from the term immediately following it: d = aₙ₊₁ − aₙ. You can verify correctness by repeating this with different pairs throughout the sequence. If every adjacent pair yields the same result, your sequence is arithmetic.

Is the sequence −12, −1, 10, 21 actually arithmetic?

Yes. The common difference is 11. Checking: −1 − (−12) = 11, 10 − (−1) = 11, and 21 − 10 = 11. All differences match, confirming this is a valid arithmetic sequence. Notice that starting with a negative number doesn't prevent arithmetic progression; consistency is what matters.

How do I compute the sum of many terms without adding individually?

Use the formula S = n/2 × (a₁ + aₙ), where n is the count of terms, a₁ is the first, and aₙ is the last. Alternatively, S = n/2 × (2a₁ + (n − 1) × d). For the sequence 2, 5, 8, 11, 14 (five terms), the sum is 5/2 × (2 + 14) = 5/2 × 16 = 40, far faster than 2 + 5 + 8 + 11 + 14.

What happens when I sum an infinite arithmetic sequence?

An infinite arithmetic sequence with a non-zero common difference always sums to infinity. Since each term either grows steadily (positive d) or shrinks indefinitely (negative d), no finite limit exists. Only special sequences like geometric progressions with a ratio between −1 and 1 can have finite infinite sums.

How do I check if a list of numbers forms an arithmetic sequence?

Calculate the difference between each consecutive pair. If all differences are identical, the sequence is arithmetic. For example, with 5, 10, 15, 20: 10 − 5 = 5, 15 − 10 = 5, 20 − 15 = 5. All equal, so it's arithmetic. If any difference differs, the sequence doesn't follow an arithmetic pattern.

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