Sequences and series formulas

A sequence is a list of numbers that follows a rule; a series is what you get when you add them up. These formulas find any term without listing the ones before it, add up long runs of terms in one step, and tell you when an endless sum still has a finite total.

Printable cheat sheet Formula finder

Arithmetic sequence (n-th term)

aₙ = a₁ + (n − 1)d

Finds any term of a sequence that goes up or down by the same amount each step, such as 3, 7, 11, 15, …

Arithmetic series (sum)

Sₙ = n/2(2a₁ + (n − 1)d)

Adds up the first n terms of an arithmetic sequence in one step.

Geometric sequence (n-th term)

aₙ = a₁r^( n − 1)

Finds any term of a sequence that is multiplied by the same ratio each step, such as 3, 6, 12, 24, …

Geometric series (sum)

Sₙ = (a₁(1 − rⁿ))/(1 − r), r ≠ 1

Adds up the first n terms of a geometric sequence; when r = 1 the sum is simply $n a_{1}$ .

Infinite geometric series (sum to infinity)

S = a₁/(1 − r), |r| < 1

The total of a never-ending geometric series whose terms shrink, such as 1 + ½ + ¼ + … = 2.

Sum of the first n whole numbers

1 + 2 + 3 + ⋯ + n = (n(n + 1))/2

Adds 1 + 2 + … + n instantly; the answers 1, 3, 6, 10, 15, … are the triangular numbers.

Sum of the first n squares

1² + 2² + ⋯ + n² = (n(n + 1)(2n + 1))/6

Adds $1² + 2² + ⋯ + n²$ in one step; it is always a whole number even though there is a division by 6.