Sequences Overview

Sequence A sequence is a pattern of an ordered list of elements (numbers, figures in a picture, letters etc.) A sequence is a function whose domain is a subset of integers (the natural #s) A term in the sequence is an individual item / element of the list. Examples: 3, 5, 7, 9… J, F, M, A, M, J, J, The “…” indicates that the pattern continues. Infinite sequences continue on forever Finite sequences terminate

Term Notation We use subscript to identify each term of the sequence starting at 1: a1 is used to identify the 1st term in the sequence. a12 is used to identify the 12th term in the sequence. an is used to identify the nth term in the sequence. Example sequence 1, 2, 4, 8, 16, 32 a1, a2, a3, a4, a5, a6, … So a1 = 1, a2 = 2, a3 = 4, a4 = 8 etc. Mean Same Thing

Types of sequences: Arithmetic Sequence In an Arithmetic Sequence the difference between one term and the next term is a constant. Each term is found by adding some constant value each time always think add, add a positive or add a negative (not subtract) The common difference is the constant number you add each time and is usually represented by the variable d For example, Describe: 1, 4, 7, 10, 13, 16, 19, 22, 25, … (you are adding 3 each time) Arithmetic sequence common difference of 3 or d=3 Made by rule (formula) an = 3n – 2. 1.5, .75, 0, -.75, -1.5, -2.25, … (you are adding -.75 each time) Arithmetic sequence common difference of -.75 or d=-.75 Made by rule (formula) an = -.75n + 2.25

Arithmetic Sequence For each sequence, determine if it is arithmetic, and find the common difference. -3, -6, -9, -12, … 1.1, 2.2, 3.3, 4.4, … 41, 32, 23, 14, 5, … 1, 2, 4, 8, 16, 32, … Arithmetic, d = -3 Arithmetic, d = 1.1 Arithmetic, d = -9 Not an arithmetic sequence.

Types of sequences: Geometric Sequence In a Geometric Sequence the ratio between one term and the next term is a constant Each term is found by multiplying the pervious term by a constant. always think multiply, multiply by an integer or by a fraction (not divide) The common ratio is the constant number you multiply by each time and is usually represented by the variable r For example, Describe: 2, 4, 8, 16, 32, 64, 128, … (you are multiplying by 2) Geometric sequence Common ratio of 2 or r=2 27, 9, 3, 1, 1/3, 1/9, 1/27 , … (you are multiplying by 1/3 ) Geometric sequence Common ratio of 1/3 or r= 1/3

Geometric Sequence For each sequence, determine if it is geometric, and find the common ratio. 2, 8, 32, 128, … 1, 10, 100, 1000, … 1, -1, 1, -1, … 20, 16, 12, 8, 4, … Geometric, r = 4 Geometric, r = 10 Geometric, r = -1 Not a geometric sequence.

write a Sequence / Find a term To write terms of a sequence: plug in the term number, n, as your input, and evaluate to find the term, an, as an output. (it’s just a function!) Example 1: Example 2: A sequence generated by the formula an = 6n – 4. Generate the first 5 terms of the sequence. a1 = 6(1) – 4 a2 = 6(2) – 4 a3 = 6(3) – 4 a4 = 6(4) – 4 a5 = 6(5) – 4 The rule is: an = 3n + 1 Find a100 a100 = 3(100) + 1 a100 = 301 2, 8, 14, 20, 26

Sequences as functions A sequence is a function whose domain is a subset of integers domain is the set of natural numbers remember natural numbers are the positive integers (whole numbers 1 and greater) the term numbers are the domain (the input) the terms of the sequence are the elements in the range (the outputs) ALL ARITHMETIC SEQUENCES ARE LINEAR FUNCTIONS because they have a constant rate of change (the common difference) Geometric Sequences are other types of functions, they often are exponential (when the common ratio is positive), but can be other kinds of functions as well (when the common ratio is negative)