SEQUENCES Unit Standard 5248 Fibonacci and rabbits 1

SEQUENCES Unit Standard 5248

Fibonacci and rabbits 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … 1/1 = 1 13/8 = 1. 625 2/1 = 2 21/13 = 1. 6153 3/2 = 1. 5 34/21 = 1. 61904 5/3 = 1. 666. . 55/34 = 1. 61764 8/5 = 1. 6 89/55 = 1. 618181 Golden ratio = 1. 618033989…

Starter ¢ Write down the next three terms in the following sequences: 1) 2) 3) 4) 5) 6) 2, 4, 6, 8, …, …, …. 3, 6, 12, 24, …, …, … 12, 7, 2, -3, …, …, … 8, 4, 2, 1, …, …, … 1, 3, 6, 10, …, …, … 2, -2, …, …, …

Definition ¢ Simply speaking a sequence is an ordered list of numbers. ¢ Every sequence has a first number, a second number, etc ¢ For most sequences there is a rule which can tell us what each number should be. ¢ The numbers in a sequence are known as terms.

Notation ¢ Any term in a sequence is written as tn, where the n refers to the position in the sequence. Eg: - ¢ t 3 is the third term, t 5 is the fifth term The first term is also referred to as a ie a = t 1

Arithmetic Sequences ¢ ¢ An ARITHMETIC sequence is one where there is a common difference between terms. Examples: a) b) c) d) e) In each example, can you 1, 2, 3, 4, …. . see what the common 3, 5, 7, 9, …… difference is? 10, 15, 20, 25, …. . 0, 0. 5, 1, 1. 5, 2, 2. 5, …. . 3, 0, -3, -6, ……

Notation for an Arithmetic Sequence In an arithmetic sequence the common difference is d. ¢ Examples: ¢ Ø Ø 1, 4, 7, 10, …. a=1, d=3 10, 8, 6, 4, …. a=10, d=-2 5, 11, 17, 23, …. a=5, d=6 1. 73, 1. 87, 2. 01, 2. 15, a=1. 73, d=0. 14

Arithmetic sequence examples ¢ i. iii. iv. v. Given a and d list the first 5 terms. a=1, d=4 a=7, d=-6 a=-95, d=3 a=13, d=11 a=1. 5, d=0. 75 i. iii. iv. v. 1, 5, 9, 13, 17 7, 1, -5, -11, -17 -95, -92, -89, -86, -83 13, 24, 35, 46, 57 1. 5, 2. 25, 3. 0, 3. 75, 4. 5

A formula for the terms of an Arithmetic sequence. ¢ ¢ The first term of an arithmetic sequence is : a To get the second term add on the d. t 2 = a + d To get the third term add another d. t 3 = a + d = a + 2 d The fourth term will be. . t 4 = t 3 +d = a + 2 d + d = a + 3 d

General term for an Arithmetic Sequence ¢ Any term in an arithmetic sequence can be found using the formula: tn = a + (n – 1)d ¢ ¢ Example: If a = 5 and d = 2 then t 100 = 5 + (100 -1). 2 = 203 tn is known as the general term or the nth term of the sequence.

Practice Examples ¢ Theta Maths (Green) Do Page 105 Exercise 13. 1 q 2, q 3, q 5, q 10, q 11