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Section 9.6 Sequences Def: A sequence is a list of items occurring in a specified order. Items may be numbers, letters, objects, movements, etc. Def: A sequence is a list of items occurring in a specified order. Items may be numbers, letters, objects, movements, etc. Examples: • clap clap stomp clap clap stomp clap clap stomp Def: A sequence is a list of items occurring in a specified order. Items may be numbers, letters, objects, movements, etc. Examples: • clap clap stomp clap clap stomp clap clap stomp A A B A A B A A B Def: A sequence is a list of items occurring in a specified order. Items may be numbers, letters, objects, movements, etc. Examples: • clap clap stomp clap clap stomp clap clap stomp A A B A A B A A B • 1, 2, 3, 4, 5, 6, 7, …. Def: A sequence is a list of items occurring in a specified order. Items may be numbers, letters, objects, movements, etc. Examples: • clap clap stomp clap clap stomp clap clap stomp A A B A A B A A B • 1, 2, 3, 4, 5, 6, 7, …. • Def: A sequence is a list of items occurring in a specified order. Items may be numbers, letters, objects, movements, etc. Examples: • clap clap stomp clap clap stomp clap clap stomp A A B A A B A A B • 1, 2, 3, 4, 5, 6, 7, …. • • …… 3 6 9 Repeating Patterns Each sequence is made up of a unit that is repeated a certain number of times or infinitely. We consider the unit to be the smallest repeated portion. Repeating Patterns Each sequence is made up of a unit that is repeated a certain number of times or infinitely. We consider the unit to be the smallest repeated portion. Repeating Patterns Each sequence is made up of a unit that is repeated a certain number of times or infinitely. We consider the unit to be the smallest repeated portion. …… Repeating Patterns Each sequence is made up of a unit that is repeated a certain number of times or infinitely. We consider the unit to be the smallest repeated portion. …… Repeated Patterns Example Problem Ex 1: What is the 75th object in the following sequence? …… Growing Patterns: Def: An arithmetic sequence begins with any number as the 1st entry, but each subsequent entry is obtained by adding or subtracting a particular fixed number to/from the previous entry. Growing Patterns: Def: An arithmetic sequence begins with any number as the 1st entry, but each subsequent entry is obtained by adding or subtracting a particular fixed number to/from the previous entry. Ex’s: 1 2 3 4 5 6 …. Growing Patterns: Def: An arithmetic sequence begins with any number as the 1st entry, but each subsequent entry is obtained by adding or subtracting a particular fixed number to/from the previous entry. Ex’s: 1 7 2 3 4 5 11 15 19 23 …. 6 …. Growing Patterns: Def: An arithmetic sequence begins with any number as the 1st entry, but each subsequent entry is obtained by adding or subtracting a particular fixed number to/from the previous entry. Ex’s: 1 2 3 4 5 7 11 15 19 23 …. 4 1 -2 -5 …. 6 …. Arithmetic Sequences • Question: How do we find the Nth entry in the sequence for some whole number N? Arithmetic Sequences • Question: How do we find the Nth entry in the sequence for some whole number N? • See Activity 9Z Arithmetic Sequences In general, the Nth term of an arithmetic sequence is (increase amount) x N + (0th entry) where the 0th entry is found by subtracting the increase amount from the 1st entry. More Growing Patterns • Def: A geometric sequence starts with any number as the 1st entry, and then each subsequent entry is obtained by multiplying or dividing by some fixed number. This fixed number, when using multiplication, is called the ratio. More Growing Patterns • Def: A geometric sequence starts with any number as the 1st entry, and then each subsequent entry is obtained by multiplying or dividing by some fixed number. This fixed number, when using multiplication, is called the ratio. • Ex’s: 3 9 27 81 ….. More Growing Patterns • Def: A geometric sequence starts with any number as the 1st entry, and then each subsequent entry is obtained by multiplying or dividing by some fixed number. This fixed number, when using multiplication, is called the ratio. • Ex’s: 3 9 27 81 ….. 1 1/2 1/4 1/8 ….. Geometric Sequences • Ex2: The amount of money in a savings account after N years of an initial deposit of $1000 that earns 4% interest annually can be viewed by the terms of the following geometric sequence: 1000 1040 1081.6 1124.87…… where the ratio of the sequence is 1.04. What is the value of the account after 6 years? Geometric Sequences For a general geometric sequence, the Nth entry is ratio 𝑁 ∙ (0th entry) where the 0th entry, if not given, can be found by dividing the 1st entry by the ratio. Other Sequences • Fibonacci sequence: starts with 1 as the 1st and 2nd entries, with each subsequent entry being found by adding its previous two entries 1 1 2 3 5 8 13 21 ….. Other Sequences • Fibonacci sequence: starts with 1 as the 1st and 2nd entries, with each subsequent entry being found by adding its previous two entries 1 1 2 3 5 8 13 21 ….. • Quadratic sequence: each entry is given by a quadratic expression • Ex: 𝑁 2 + 2𝑁 − 3 0 5 12 21 32 ….. Other Sequences • Fibonacci sequence: starts with 1 as the 1st and 2nd entries, with each subsequent entry being found by adding its previous two entries 1 1 2 3 5 8 13 21 ….. • Quadratic sequence: each entry is given by a quadratic expression • Ex: 𝑁 2 + 2𝑁 − 3 0 5 12 21 32 ….. • Non-repeating and non-growing: 3 1 4 1 5 9 2 6 5 3 5 9 …..