Arithmetic and Geometric Sequence Calculator
Find the nth term or partial sum of arithmetic and geometric sequences instantly. Use it when solving series problems, verifying patterns, or preparing for algebra and precalculus exams.
About this calculator
An arithmetic sequence grows by adding a constant difference d each step, so its nth term is aₙ = a₁ + (n − 1)d and its partial sum is Sₙ = (n/2)(2a₁ + (n − 1)d). A geometric sequence multiplies by a constant ratio r each step, giving aₙ = a₁ · rⁿ⁻¹ and partial sum Sₙ = a₁(rⁿ − 1)/(r − 1) when r ≠ 1. Choosing the right formula depends on whether consecutive terms differ by addition (arithmetic) or multiplication (geometric). These formulas appear in finance (compound interest), physics (projectile motion), and computer science (algorithmic complexity). Understanding both types builds the foundation for calculus series and convergence analysis.
How to use
Suppose you have an arithmetic sequence with first term a₁ = 3, common difference d = 5, and you want the 8th term and the sum of the first 8 terms. Nth term: a₈ = 3 + (8 − 1) × 5 = 3 + 35 = 38. Partial sum: S₈ = (8/2) × (2 × 3 + (8 − 1) × 5) = 4 × (6 + 35) = 4 × 41 = 164. Enter Sequence Type = Arithmetic, First Term = 3, Common Difference = 5, Term Number = 8, then select nth Term or Partial Sum to see either result instantly.
Frequently asked questions
What is the difference between an arithmetic and a geometric sequence?
In an arithmetic sequence, each term is obtained by adding a fixed common difference to the previous term, for example 2, 5, 8, 11 (d = 3). In a geometric sequence, each term is obtained by multiplying the previous term by a fixed common ratio, for example 2, 6, 18, 54 (r = 3). The distinction matters because the two types have entirely different formulas for their nth term and their partial sums. Mixing them up leads to wrong answers, so always identify the pattern first by checking whether the difference or the ratio between consecutive terms is constant.
How do you find the sum of a geometric series when the ratio is less than 1?
When |r| < 1, the partial sum formula Sₙ = a₁(rⁿ − 1)/(r − 1) still applies for a finite number of terms. Additionally, as n → ∞ the infinite geometric series converges to S = a₁/(1 − r) because rⁿ approaches zero. For example, with a₁ = 10 and r = 0.5, the infinite sum is 10/(1 − 0.5) = 20. This convergence property is fundamental in probability, economics (perpetuities), and signal processing. If |r| ≥ 1, the infinite series diverges and only finite partial sums are meaningful.
When should I use the partial sum formula instead of the nth term formula?
Use the nth term formula when you only need a specific term's value, such as 'what is the 20th payment in a savings plan?' Use the partial sum formula when you need the total accumulated value over n steps, such as 'how much will I have saved after 20 months?' In finance, partial sums model total interest earned or total distance traveled under uniform acceleration. Many exam problems ask for both, so it is worth computing each separately. This calculator lets you switch between the two goals with a single dropdown, saving manual re-entry.