Arithmetic Sequence Calculator

An arithmetic sequence changes by the same additive amount from one term to the next. Once the first term and common difference are known, every term is determined. This calculator also works backward from two indexed terms, so you can recover the underlying sequence instead of needing a₁ and d in advance.

How to use the Arithmetic Sequence Calculator

If you know the first term and common difference, select First term + common difference. Enter a₁, d, and the term number n you want. The calculator returns the requested term, the explicit nth-term rule, the sum of the first n terms, and a preview of the sequence.

If your problem gives two terms such as a₄ = 17 and a₁₃ = 53, choose Two known terms. Enter each index with its value, then choose a target n. The calculator first finds the common difference from the change in term value divided by the change in index. It then reconstructs a₁ and solves the entire sequence.

Term values may be integers, decimals, or fractions such as 3/2 or −7/4. Indices must be positive whole numbers because a₁ is treated as the first term.

What is an arithmetic sequence?

An arithmetic sequence is a sequence in which the difference between consecutive terms is constant. That fixed value is called the common difference, usually written d.

For 5, 9, 13, 17, ... the common difference is 4. For 20, 14, 8, 2, ... it is −6. A constant sequence such as 7, 7, 7, 7, ... is also arithmetic, with d = 0.

The defining feature is repeated addition, not whether the terms increase. A negative common difference simply means the same negative amount is added each time.

The nth-term formula

Starting at a₁, you add d once to reach a₂, twice to reach a₃, and n−1 times to reach aₙ. Therefore:

The n−1 is essential. The first term requires zero additions of d, the second requires one, and so on. A common error is to use a₁ + nd, which shifts every requested term by one common difference.

For a₁ = 3 and d = 6, the fifteenth term is a₁₅ = 3 + (15−1)6 = 87. You do not need to write all fourteen earlier transitions by hand.

How to solve an arithmetic sequence from two known terms

Suppose you know aᵢ and aⱼ at two different indices i and j. Between those positions there are j−i equal steps of size d, so:

After finding d, recover the first term with:

For a₄ = 17 and a₁₃ = 53, d = (53−17)/(13−4) = 36/9 = 4. Then a₁ = 17 − 3(4) = 5, so the sequence is 5, 9, 13, 17, ... .

The order of the two known terms does not matter if numerator and denominator use the same order. Reversing both differences changes both signs and leaves the ratio unchanged.

Arithmetic sequence vs arithmetic series

A sequence is the ordered list of terms. A series is a sum of terms from a sequence. The sum of the first n terms is usually written Sₙ.

Substituting the nth-term formula gives an equivalent version:

The first formula has a useful interpretation: in an arithmetic sequence, the average of the first and last terms equals the average term value, so multiplying that average by n gives the total.

For 8, 13, 18, ... with n = 30, a₃₀ = 153. Then S₃₀ = 30(8+153)/2 = 2,415.

Explicit form vs recursive form

An arithmetic sequence can be described recursively:

This tells you how to get the next term from the previous one. It is intuitive for building a sequence term by term.

The explicit formula aₙ = a₁ + (n−1)d is more direct when you need a distant term. To find a₁₀₀₀, the explicit rule jumps directly to it rather than computing 999 intermediate values.

Both forms describe the same sequence. The calculator shows the explicit rule because it is the most useful compact summary for arbitrary n.

Positive, negative, and zero common differences

If d > 0, the sequence increases by a constant amount. If d < 0, it decreases by a constant amount. If d = 0, all terms are equal.

The magnitude |d| measures the fixed change per index step. A larger positive d produces faster linear growth across indices; a more negative d produces faster linear decline.

Unlike a geometric sequence, an arithmetic sequence does not multiply by a fixed ratio. Adding 5 repeatedly and multiplying by 5 repeatedly create fundamentally different patterns.

Fractions, decimals, and exact sequence values

Arithmetic sequences do not have to use whole numbers. A first term of 1/3 and common difference 1/6 produces 1/3, 1/2, 2/3, 5/6, ... . Fractional common differences are perfectly valid.

SonoCalculator preserves rational values exactly when you enter integers, terminating decimals, or fractions. This matters because a value such as 1/3 is not exactly representable as a finite decimal. Keeping the fraction avoids introducing rounding into later terms and sums.

The decimal-display control provides a supporting approximation when useful, but it does not replace an exact fraction in the algebra.

The connection between arithmetic sequences and linear functions

The explicit formula can be expanded:

This has the same form as a linear function y = mx + b. If the horizontal coordinate is the integer index n, then the slope is d. That is why a plot of arithmetic-sequence terms lies on a straight line at equally spaced integer x-values.

The connection also explains the two-known-term formula. Finding d from two indexed terms is the same rate-of-change idea as finding a slope from two points: change in term value divided by change in index.

But an arithmetic sequence is normally defined only at discrete integer indices, while a linear function may be defined for a continuous range of x-values.

Worked arithmetic-sequence examples

Common arithmetic-sequence mistakes

Using n instead of n−1. There are n−1 intervals from the first term to the nth term.

Confusing sequence and series. aₙ is one term; Sₙ is the sum of multiple terms.

Changing the subtraction order inconsistently. When finding d from two terms, reverse both the value difference and index difference together.

Assuming d must be positive. Negative and zero common differences are valid.

Using a ratio test instead of a difference test. Constant differences indicate arithmetic behavior; constant ratios indicate geometric behavior.

Rounding fractions too early. Preserve exact values through the formula when possible, especially when later terms or sums are required.

Using index 0 without adjusting the formula. This calculator uses the standard a₁-first convention. A sequence indexed from 0 needs a correspondingly shifted expression.

Frequently asked questions

How do I find the nth term of an arithmetic sequence?

Use aₙ = a₁ + (n−1)d, where a₁ is the first term and d is the common difference.

How do I find the common difference?

Subtract one term from the next. If two nonconsecutive indexed terms are known, divide their value difference by their index difference.

How do I find the sum of the first n terms?

Use Sₙ = n(a₁+aₙ)/2 or the equivalent formula Sₙ = n[2a₁+(n−1)d]/2.

Can an arithmetic sequence decrease?

Yes. A negative common difference produces a decreasing arithmetic sequence.

Is a constant sequence arithmetic?

Yes. Its common difference is 0.

Can an arithmetic sequence contain fractions or decimals?

Yes. The first term and common difference can be any suitable real values; this calculator preserves rational entries exactly when practical.

Is an arithmetic sequence the same as a linear function?

They share a linear formula, but a sequence is ordinarily indexed at discrete integers, whereas a linear function may be evaluated continuously.

Can two known terms determine the entire sequence?

Yes, provided their indices are different. Two indexed terms determine the common difference and therefore the first term and explicit rule.