Sequence & Series Calculator
Find the nth term and the sum of an arithmetic or geometric sequence. Shows the formulas, every step, and a plot of the terms.
About this calculator
Pick arithmetic (constant difference d) or geometric (constant ratio r). Enter the first term a₁, the difference or ratio, and how many terms n. The calculator returns the nth term aₙ and the partial sum Sₙ with full working.
Arithmetic vs. geometric sequences
An arithmetic sequence adds a fixed difference d between terms, so the nth term is aₙ = a₁ + (n − 1)d and the sum of the first n terms is Sₙ = n/2 · (2a₁ + (n − 1)d). A geometric sequence multiplies by a fixed ratio r, so aₙ = a₁ · r^(n−1) and the sum is Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1. When r = 1 every term equals a₁, so the sum is simply n · a₁. Plotting the first terms makes the shape obvious: arithmetic terms lie on a straight line, while geometric terms curve (growing fast when |r| > 1, shrinking toward zero when |r| < 1).