#separator:tab #html:false #tags column:4 Sequence An ordered list of numbers following a rule math-sequences-series Term of a sequence One number in the list math-sequences-series Arithmetic sequence Each term is found by adding the same number math-sequences-series Common difference The number added each time in an arithmetic sequence math-sequences-series Geometric sequence Each term is found by multiplying by the same number math-sequences-series Common ratio The number multiplied each time in a geometric sequence math-sequences-series Explicit formula, arithmetic aₙ = a₁ + (n − 1)d math-sequences-series Explicit formula, geometric aₙ = a₁ · rⁿ⁻¹ math-sequences-series Recursive formula Gives each term from the term before it math-sequences-series Sum of an arithmetic series Sₙ = n(a₁ + aₙ)/2 math-sequences-series Sum of a finite geometric series Sₙ = a₁(1 − rⁿ)/(1 − r) math-sequences-series Sum of an infinite geometric series S = a₁/(1 − r), when |r| < 1 math-sequences-series Next term: 2, 4, 6, 8, ... 10 (add 2) math-sequences-series Next term: 5, 10, 15, 20, ... 25 (add 5) math-sequences-series Next term: 3, 6, 12, 24, ... 48 (multiply by 2) math-sequences-series Next term: 1, 4, 9, 16, ... 25 (squares) math-sequences-series Next term: 1, 3, 6, 10, ... 15 (triangular numbers, add 5) math-sequences-series Next term: 100, 90, 80, ... 70 (subtract 10) math-sequences-series Next term: 81, 27, 9, 3, ... 1 (divide by 3) math-sequences-series Next term: 1, 1, 2, 3, 5, 8, ... 13 (Fibonacci) math-sequences-series Next term: 2, 6, 18, 54, ... 162 (multiply by 3) math-sequences-series Next term: 1, 2, 4, 7, 11, ... 16 (differences grow by 1) math-sequences-series Next term: 1, 8, 27, 64, ... 125 (cubes) math-sequences-series Next term: 2, 3, 5, 7, 11, ... 13 (primes) math-sequences-series 10th term of 3, 7, 11, ... 39 math-sequences-series 20th term of 5, 8, 11, ... 62 math-sequences-series 8th term of 2, 4, 8, ... 256 math-sequences-series Common difference of 7, 12, 17, ... 5 math-sequences-series Common ratio of 5, 15, 45, ... 3 math-sequences-series Common ratio of 16, 8, 4, ... 1/2 math-sequences-series Sum of 1 + 2 + ... + 10 55 math-sequences-series Sum of 1 + 2 + ... + 100 5,050 math-sequences-series Sum of the first 5 odd numbers 25 math-sequences-series Sum of the first 10 even numbers 110 math-sequences-series Sum of 1 + 1/2 + 1/4 + 1/8 + ... forever 2 math-sequences-series Fibonacci rule Each term is the sum of the two before it math-sequences-series Fibonacci numbers up to 144 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 math-sequences-series Ratio of consecutive Fibonacci numbers Approaches the golden ratio, about 1.618 math-sequences-series Triangular numbers formula n(n + 1)/2 math-sequences-series Square numbers formula n² math-sequences-series Pentagonal number idea n(3n − 1)/2: 1, 5, 12, 22, ... math-sequences-series Pascal's triangle rows 1; 1 1; 1 2 1; 1 3 3 1; 1 4 6 4 1 math-sequences-series Row sums of Pascal's triangle Powers of 2 math-sequences-series Sigma notation Σ means add up the terms for the given range math-sequences-series Σ from k=1 to 4 of k 10 math-sequences-series Σ from k=1 to 3 of k² 14 math-sequences-series Σ from k=1 to 5 of 2 10 math-sequences-series Pattern Something that repeats or changes by a rule math-sequences-series Growth pattern A pattern in which a number or shape grows by a rule each step math-sequences-series Limit of a sequence The value the terms get close to math-sequences-series Converge The terms or sum settle toward a number math-sequences-series Diverge The terms or sum do not settle toward a number math-sequences-series Harmonic series 1 + 1/2 + 1/3 + 1/4 + ... ; it grows without bound math-sequences-series Arithmetic mean of two numbers Their average math-sequences-series Geometric mean of two numbers The square root of their product math-sequences-series Geometric mean of 4 and 9 6 math-sequences-series Compound interest as a sequence A geometric sequence, multiplying by (1 + r) each period math-sequences-series