Sequences & Series
Find the nth term and the sum of arithmetic and geometric sequences, with the formula and substitution shown. Try “sum of the first 20 terms of 3, 7, 11, …”.
Try an example:
How the Sequences & Series works
- Decide whether there is a common difference d (arithmetic) or common ratio r (geometric).
- Use aₙ = a₁ + (n − 1)d or aₙ = a₁rⁿ⁻¹ for the nth term.
- Use Sₙ = n/2 (a₁ + aₙ) or Sₙ = a₁(1 − rⁿ) ÷ (1 − r) for the sum.
Frequently asked questions
How can I tell arithmetic from geometric?
Arithmetic sequences add the same amount each time; geometric sequences multiply by the same amount.
When does an infinite geometric series converge?
When |r| < 1, and then the sum is a₁ ÷ (1 − r).
More Algebra calculators
Linear Equation SolverSolve one-variable equations with balancing steps.
Quadratic Equation SolverFactoring, completing the square and the quadratic formula.
System of EquationsSubstitution, elimination and matrix methods.
Inequality SolverLinear, quadratic and absolute-value inequalities with number lines.
Factoring & ExpandingFactor or expand polynomials and expressions.
Polynomial DivisionLong division and synthetic division, remainder theorem.
MathMate can make mistakes. Please double-check important steps and calculations.