Arithmetic Sequence Formulas
Use arithmetic sequence formulas to find the common difference, nth term, recursive rule, and sum of a finite arithmetic series.
An arithmetic sequence changes by a constant difference. These formulas describe its pattern, individual terms, and partial sums.
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5 formulas, different jobs
An arithmetic sequence changes by a constant difference. These formulas describe its pattern, individual terms, and partial sums.
Common Difference
Use it to check the fixed change between consecutive terms.
Explicit Formula
Use it to find any term directly from its position.
Recursive Formula
Use it to generate each term from the previous term.
Finite Sum Using First and Last Terms
Use it when the first term, last term, and number of terms are known.
Finite Sum Using the Difference
Use it when the last term is unknown but the common difference is known.
Find a term and the total
For the sequence 3, 9, 15, 21, …, find the tenth term and the sum of the first ten terms.
- 1 · Difference
- 2 · Explicit rule
- 3 · Tenth term
- 4 · Sum
Common mistakes
- Using n instead of n − 1 in the explicit formula.
- Calling a sequence arithmetic when its consecutive differences are not constant.
- Confusing aₙ, one term, with Sₙ, the sum of several terms.
See which formulas your problem uses
Enter a problem and get a step-by-step solution with the formulas and theorems identified.
Calculators and related formula guides
Algebra Calculator
Solve sequence equations and simplify their formulas.
Equation Calculator
Find unknown terms and parameters in a sequence.
Geometric sequences
Use geometric sequence formulas to find the common ratio, nth term, recursive rule, finite sum, and convergent infinite sum.
Slope formulas
Compare the slope formula, point-slope form, slope-intercept form, standard form, and special horizontal and vertical lines.
Arithmetic sequences FAQ
What makes a sequence arithmetic?
The difference between every pair of consecutive terms must be the same constant d.
What is the difference between a sequence and a series?
A sequence is an ordered list of terms. A series is the sum of those terms.