Sequences and Series

A list of 4 common types of sequences: Arithmetic Sequences Geometric Sequences Harmonic Sequences Fibonacci Numbers

Edited at 2020-09-08 01:10:56

Geometric

Geometric Sequence

A geometric sequence is a sequence where a common ratio is multiplied to each consecutive term.

Examples : 1, 2, 4, 8, 16 r = 2 1, 3, 9, 27, 81 r = 3 1, 5, 25, 125, 625 r = 5

Formula : An = A1r^n-1

Geometric Series

A geometric series is the sum of all the terms of a geometric sequence.

Examples : 1, 2, 4, 8, 16 S5 = 31 1, 3, 9, 27, 81 S5 = 121 1, 5, 25, 125, 625 S5 = 781

Formula : (Finite) Sn = A1 (1-r^n) (-------) ( 1-r ) or Sn = A1 (r^n-1) (-------) ( r-1 ) Formula : (Infinite) Sn = A1/r-1

Fibonacci Sequence

A fibonacci sequence is a sequence where the previous term is added to the next term.

Example : 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377

Harmonic Sequence

A harmonic sequence is a sequence where the reciprocals of the terms form an arithmetic sequence.

Examples : 1, 1/2, 1/3, 1/4, 1/5 1, 1/16, 1/33, 1/50, 1/67, 1/84 1, -1/3, -1/7, -1/11, -1/15

Arithmetic

Arithmetic Sequence

An arithmetic sequence is a sequence where a common difference is added to each consecutive term.

Examples : 1, 2, 3, 4, 5 d = 1 16, 33, 50, 67, 84 d = 17 1, -3, -7, -11, -15 d = -4

Formula : An = A1 + (n-1) d

Arithmetic Series

An arithmetic series is the sum of all the terms of an arithmetic sequence.

Examples : 1, 2, 3, 4, 5 S5 = 15 16, 33, 50, 67, 84 S5 = 250 1, -3, -7, -11, -15 S5 = -35

Formula : Sn = n/2 (A1+An)