Arithmetic Sequence Calculator
FreeCalculate the nth term and sum of an arithmetic sequence, with a sequence preview, bar chart, and reverse lookup for a target value.
Runs entirely in your browser.
Step-by-step
Which term has a given value?
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What this tool does
This tool calculates the nth term and the sum of the first n terms of an arithmetic sequence, right in your browser, given the first term and common difference, with a sequence preview, a bar chart, and a reverse lookup to find which term holds a given value.
How It Works
Given a first term a₁ and a common difference d, the nth term is aₙ = a₁ + (n − 1)d, since d is added once less than the term's position. The sum of the first n terms is Sₙ = n(2a₁ + (n − 1)d) / 2, equivalent to Sₙ = n(a₁ + aₙ) / 2 once aₙ is known, both come from pairing terms from opposite ends of the sequence. To find which term equals a target value, the nth-term formula is solved in reverse: n = (target − a₁) / d + 1, and the target is only actually in the sequence if that comes out to a positive whole number.
Problems it solves
- Finding a specific term far into a sequence without listing out every term by hand.
- Computing the total of many evenly-spaced values, such as a savings plan with fixed increases.
- Checking whether a specific number belongs to a given arithmetic sequence, and at which position.
- Visualizing how a sequence grows, shrinks, or stays flat with a bar chart.
Frequently asked questions
›What is an arithmetic sequence?
It is a list of numbers where each term after the first is found by adding a fixed amount, the common difference d, to the previous term. If d is positive the sequence increases, if d is negative it decreases, and if d is zero every term is the same.
›How is the nth term formula derived?
Starting from a₁, you add d exactly (n − 1) times to reach the nth term, since the first term needs zero additions, the second needs one, and so on. That gives aₙ = a₁ + (n − 1)d directly.
›Why does the sum formula use n(a₁ + aₙ)/2?
Pairing the first term with the last, the second with the second-to-last, and so on, each pair adds up to the same total, a₁ + aₙ. There are n/2 such pairs, which is where the classic "Gauss trick" for summing a sequence comes from.
›What does "which term has a given value" do?
It solves the nth term formula in reverse for n, given a target value: n = (target − a₁) / d + 1. If that comes out to a positive whole number, the target is in the sequence at that position, otherwise it never appears.
›Can the common difference or first term be negative or a decimal?
Yes, both a₁ and d can be any real number, positive, negative, zero, or a decimal, the formulas work identically regardless.
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