What is happening here? 1, 1, 2, 3, 5, 8 What is after 8? What is the 10 th number?

Slides:

Advertisements

Similar presentations

11.5 Recursive Rules for Sequences p Explicit Rule A function based on a term’s position, n, in a sequence. All the rules for the nth term that.

Advertisements

Warm up 1. Determine if the sequence is arithmetic. If it is, find the common difference. 35, 32, 29, 26, Given the first term and the common difference.

RECURSIVE FORMULAS In addition to level 3.0 and above and beyond what was taught in class, the student may: · Make connection with other concepts.

Copyright © 2007 Pearson Education, Inc. Slide 8-1 Warm-Up Find the next term in the sequence: 1, 1, 2, 6, 24, 120,…

Number Patterns. What are they? Number patterns are a non-ending list of numbers that follow some sort of pattern from on number to the next.

7.5 Use Recursive Rules with Sequences and Functions

Arithmetic Sequences and Series

Section 11.2 Arithmetic Sequences

1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 10 Further Topics in Algebra.

Arithmetic Sequences A sequence in which each term after the first is obtained by adding a fixed number to the previous term is an arithmetic sequence.

9.2 Arithmetic Sequence and Partial Sum Common Difference Finite Sum.

Arithmetic Sequences and Series. A sequence is arithmetic if each term – the previous term = d where d is a constant e.g. For the sequence d = 2 nd term.

Sequences and Series It’s all in Section 9.4a!!!.

Aim: What are the arithmetic series and geometric series? Do Now: Find the sum of each of the following sequences: a)

1 © 2010 Pearson Education, Inc. All rights reserved 10.1 DEFINITION OF A SEQUENCE An infinite sequence is a function whose domain is the set of positive.

Arithmetic Sequences. A mathematical model for the average annual salaries of major league baseball players generates the following data. 1,438,0001,347,0001,256,0001,165,0001,074,000983,000892,000801,000.

12-1 Arithmetic Sequences and Series. Sequence- A function whose domain is a set of natural numbers Arithmetic sequences: a sequences in which the terms.

THE BEST CLASS EVER…ERRR…. PRE-CALCULUS Chapter 13 Final Exam Review.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Explicit, Summative, and Recursive

Copyright © Cengage Learning. All rights reserved.

What are two types of Sequences?

12.1 Sequences and Series ©2001 by R. Villar All Rights Reserved.

Arithmetic Sequences Standard: M8A3 e. Use tables to describe sequences recursively and with a formula in closed form.

Notes 9.4 – Sequences and Series. I. Sequences A.) A progression of numbers in a pattern. 1.) FINITE – A set number of terms 2.) INFINITE – Continues.

Sequences & Series. Sequences  A sequence is a function whose domain is the set of all positive integers.  The first term of a sequences is denoted.

Review of Sequences and Series.  Find the explicit and recursive formulas for the sequence:  -4, 1, 6, 11, 16, ….

1 1 OBJECTIVE At the end of this topic you should be able to Define sequences and series Understand finite and infinite sequence,

Sigma Notation A compact way of defining a series A series is the sum of a sequence.

Sequences, Series, and Sigma Notation. Find the next four terms of the following sequences 2, 7, 12, 17, … 2, 5, 10, 17, … 32, 16, 8, 4, …

Sequences and Summations Section 2.4. Section Summary Sequences. – Examples: Geometric Progression, Arithmetic Progression Recurrence Relations – Example:

In this lesson you will learn another way to define a sequence — by a recursive rule. So far you have worked with explicit rules for the n th term of a.

Unit 5 – Series, Sequences, and Limits Section 5.2 – Recursive Definitions Calculator Required.

Section 9-4 Sequences and Series.

Geometric Sequences & Series This chapter focuses on how to use find terms of a geometric sequence or series, find the sum of finite and infinite geometric.

Algebra II Chapter : Use Recursive Rules with Sequences and Functions HW: p (4, 10, 14, 18, 20, 34)

Standard Accessed: Students will analyze sequences, find sums of series, and use recursive rules.

Infinite Geometric Series Recursion & Special Sequences Definitions & Equations Writing & Solving Geometric Series Practice Problems.

Sequences & Series: Arithmetic, Geometric, Infinite!

11.2 & 11.3: Sequences What is now proven was once only imagined. William Blake.

4.2B Geometric Explicit and Recursive Sequences

4.2A Arithmetic Explicit and Recursive Sequences

Review of Sequences and Series

Patterns in Sequences. Number patterns Sequences of numbers can have interesting patterns. Here we list the most common patterns and how they are made.

EXAMPLE 1 Evaluate recursive rules Write the first six terms of the sequence. a. a 0 = 1, a n = a n – b. a 1 = 1, a n = 3a n – 1 SOLUTION a. a 0.

{1, 1, 2, 3, 5, 8, 13, 21,...} What is this? Fibonacci Sequence.

11.5 Recursive Rules for Sequences p What is a recursive rule for sequences? What does ! mean in math?

MDFP Mathematics and Statistics 1 ARITHMETIC Progressions.

MDFP Mathematics and Statistics 1 GEOMETRIC Progressions.

Arithmetic Sequences and Series Section Objectives Use sequence notation to find terms of any sequence Use summation notation to write sums Use.

Arithmetic and Geometric sequence and series

Sequences and Series when Given Two Terms and Not Given a1

SEQUENCES AND SERIES.

The symbol for summation is the Greek letter Sigma, S.

Sequence and Series Review Problems

Geometric Series When the terms of a geometric sequence are added, the result is a geometric series The sequence 3, 6, 12, 24, 48…gives rise to the series.

Sequences and Series.

Using Recursive Rules for Sequences

Sequences & Series.

Unit 1 Test #3 Study Guide.

10.2 Arithmetic Sequences and Series

Notes Over 11.5 Recursive Rules

Geometric Sequences and Series

Chapter 12 Review Each point your team earns is extra points added to your score on the upcoming test.

Section 12.1 Sequences and Section 12.2 Arithmetic Sequences

8.5 Using Recursive Rules with Sequences

Warm Up Write the first 4 terms of each sequence:

Lesson 6.7 Recursive Sequences

Presentation transcript:

What is happening here? 1, 1, 2, 3, 5, 8 What is after 8? What is the 10 th number?

What is happening here? 1, 1, 2, 3, 5, 8 What is after 8? What is the 10 th number?

Welcome to Core 2 - Sequences Know what the “recursive definition” of a “sequence” are. Understand how to generate the terms of a sequence from recursive definition Be able to solve problems using simultaneous equations using the above.

Sequences You are expected to understand 3 types of sequences. Recursive, Arithmetic, and geometric. Keywords Sequence: a list of numbers e.g. 1, 2, 3, 4, … Series: a sum of numbers e.g …

Terms and notation 1, 1, 2, 3, 5, 8, … First term in seq. 4 th term in seq. n th term in seq. Sometimes n is another letter, for example, r, u r. What ever n =, that’s the number term you want

Recursive Sequence 1, 2, 3, 4, 5, 6, … A recursive sequence is a rule that gives you the next term using the preceding term. You’ll be given the first term.

Recursive Sequence 1, 2, 3, 4, 5, 6, … What is happening? What are we doing each time to get to the next term? The next term is found… …using The current/preceding term… …and adding one Equation is called the recursive definition

Recursive Sequence 1, 1, 2, 3, 5, 8, … This is the Fibonacci sequence The next term is found… …using The current/preceding term… …and adding the previous term

Recursive Sequence Given that the first term, u 1, is 5, what are the first five terms in this sequence? 5, 15, 45, 135, 405

Recursive Sequence What is the recursive definition for this sequence? 28, 14, 7, 3.5, 1.75

Questions Six questions on back, have a go at them! Be sure to mark your work using the solutions

Simultaneous Equations Problems will occur that involve the use of simultaneous equations, but they will not mention it at all.

Simultaneous Equations It will be set up like this: The recursive definition of a function is u n+1 = pu n + q The first three terms are 6, -7, 19 Find the values of p and q. Unknowns in equation & first 3 terms This set up is common and requires simultaneous equations!

Simultaneous Equations u n+1 = pu n + q The first three terms are 6, -7, 19 Have a go and finish off question

Try this one out The recursive definition of a function is u n+1 = a + bu n The first three terms are 8, 10, 11… Find the values of a and b. Find u 4

Try this one out The recursive definition of a function is u n+1 = a + bu n The first three terms are 6, 11.5, 19.75… Find the values of a and b. Find u 4

Try this one out The recursive definition of a function is u n = pu n-1 + q The first three terms are 2, -14, 50… Find the values of p and q. Find u 4 & u 5

Independent Study Exercise 2A p244 (solutions p430)