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Sequences and series cheat sheet

All 7 formulas on one page. Print it or save it as a PDF; it fits on 2 sides of A4 or Letter.

Arithmetic sequence (n-th term)

aₙ = a₁ + (n − 1)d

$a_{n}$ the n-th term: the one you want
$a_{1}$ the first term
$d$ the common difference: what you add each step (negative if the sequence goes down)
$n$ the position of the term (1 for the first, 2 for the second, …)

Arithmetic series (sum)

Sₙ = n/2(2a₁ + (n − 1)d)

Using the last term

Sₙ = n/2(a₁ + aₙ)

$S_{n}$ the sum of the first n terms, $S_{n}$
$a_{1}$ the first term
$d$ the common difference
$n$ how many terms are added

Geometric sequence (n-th term)

aₙ = a₁r^( n − 1)

$a_{n}$ the n-th term
$a_{1}$ the first term (not 0)
$r$ the common ratio: what each term is multiplied by
$n$ the position of the term (1, 2, 3, …)

Geometric series (sum)

Sₙ = (a₁(1 − rⁿ))/(1 − r), r ≠ 1

Equivalent form (neater when r > 1)

Sₙ = (a₁(rⁿ − 1))/(r − 1)

When r = 1

Sₙ = na₁

$S_{n}$ the sum of the first n terms, $S_{n}$
$a_{1}$ the first term (not 0)
$r$ the common ratio
$n$ how many terms are added

Infinite geometric series (sum to infinity)

S = a₁/(1 − r), |r| < 1

$S$ the sum to infinity, $S_∞$
$a_{1}$ the first term
$r$ the common ratio; it must be strictly between −1 and 1

Sum of the first n whole numbers

1 + 2 + 3 + ⋯ + n = (n(n + 1))/2

Sigma notation

∑[k = 1..n] k = (n(n + 1))/2

$S$ the sum $1 + 2 + ⋯ + n$
$n$ the last (and largest) number added

Sum of the first n squares

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

Sigma notation

∑[k = 1..n] k² = (n(n + 1)(2n + 1))/6

$S$ the sum $1² + 2² + ⋯ + n²$
$n$ the last number squared

letx.io · Sequences and series cheat sheet · 2026-10-31