Grade 11 Functions (MCR3U)

Slides:

Advertisements

Similar presentations

Choi 2012 Arithmetic Sequence A sequence like 2, 5, 8, 11,…, where the difference between consecutive terms is a constant, is called an arithmetic sequence.

Advertisements

CN College Algebra Ch. 11: Sequences 11.3: Geometric Sequences Goals: Determine if a sequence is geometric. Find a formula for a geometric sequence. Find.

Choi 2012 What is a Series? A series is the sum of terms of a sequence. The sum of the first n terms in a sequence is denoted as S n. S 5  the sum of.

Choi Geometric Sequence A sequence like 3, 9, 27, 81,…, where the ratio between consecutive terms is a constant, is called a geometric sequence. In a.

11.4 Geometric Sequences Geometric Sequences and Series geometric sequence If we start with a number, a 1, and repeatedly multiply it by some constant,

Choi Recall: What is a Series? A series is the sum of terms of a sequence. The sum of the first n terms in a sequence is denoted as S n. S 5  the sum.

5-7: The Binomial Theorem

Lesson 6.8A: The Binomial Theorem OBJECTIVES:  To evaluate a binomial coefficient  To expand a binomial raised to a power.

Binomial – two terms Expand (a + b) 2 (a + b) 3 (a + b) 4 Study each answer. Is there a pattern that we can use to simplify our expressions?

3-1 © 2011 Pearson Prentice Hall. All rights reserved Chapter 12 Sequences, Series, and the Binomial Theorem Active Learning Questions.

Pg. 601 Homework Pg. 606#1 – 6, 8, 11 – 16 # … + (2n) 2 # (3n + 1) #5 #7(3n 2 + 7n)/2 #84n – n 2 #21#23 #26 #29 #33The series.

9.3 Geometric Sequences and Series. Common Ratio In the sequence 2, 10, 50, 250, 1250, ….. Find the common ratio.

How do I Multiply Polynomials? How do I use Binomial Expansion?

Grade 10 Academic (MPM2D) Unit 3: Algebra & Quadratic Models Product of Binomials & Polynomials Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved.

Grade 10 Academic (MPM2D) Unit 1: Linear System Word Problems with Linear Systems 1 Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved.

Grade 10 Academic (MPM2D) Unit 1: Linear System Solving Linear Systems by Elimination Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved.

Grade 10 Academic (MPM2D) Unit 5: Trigonometry Solving Right Triangles

Grade 10 Academic (MPM2D) Unit 4: Quadratic Relations Discriminants of Quadratics Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved.

Grade 10 Academic (MPM2D) Unit 1: Linear System Slope and Linear Relations Mr. Choi © 2016 E. Choi – MPM2D - All Rights Reserved.

Classifying, Adding, and Subtracting Polynomials

Grade 10 Academic (MPM2D) Unit 3: Algebra & Quadratic Models Factoring Non-Simple Trinomials Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved.

Grade 10 Academic (MPM2D) Unit 4: Quadratic Relations Perfect Square Trinomials and Completing the Square Mr. Choi © 2017 E. Choi – MPM2D - All Rights.

Grade 10 Academic (MPM2D) Unit 5: Trigonometry Congruent & Similar triangles Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved.

Grade 10 Academic (MPM2D) Unit 4: Quadratic Relations Quadratic Formula Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved.

Grade 10 Academic (MPM2D) Unit 4: Quadratic Relations The Quadratic Relation (Vertex Form) - Translations Mr. Choi © 2017 E. Choi – MPM2D - All Rights.

Grade 10 Academic (MPM2D) Unit 5: Trigonometry Modelling with Similar triangles Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved.

Arithmetic Sequences and Series

Grade 10 Academic (MPM2D) Unit 3: Algebra & Quadratic Models Finite Differences Investigations Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved.

Arithmetic and Geometric Series

Grade 10 Academic (MPM2D) Unit 4: Quadratic Relations Optimization Problems Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved.

Grade 10 Academic (MPM2D) Unit 4: Quadratic Relations The Quadratic Relations (Vertex Form) – Transformations Mr. Choi © 2017 E. Choi – MPM2D - All Rights.

11.3 Geometric sequences; Geometric Series

Created by Mr. Lafferty Maths Dept.

Arithmetic and Geometric

Objectives Find terms of a geometric sequence, including geometric means. Find the sums of geometric series.

Grade 11 Functions (MCR3U)

Grade 11 Functions (MCR3U)

Classifying, Adding, and Subtracting Polynomials

How do I Multiply Polynomials? How do I use Binomial Expansion?

6.8B– Sum of a Geometric Series

Grade 11 Functions (MCR3U)

Grade 11 Functions (MCR3U)

Classifying, Adding, and Subtracting Polynomials

Grade 10 Academic (MPM2D) Unit 1: Linear System Solving First Degree Equations Mr. Choi © 2016 E. Choi – MPM2D - All Rights Reserved.

Grade 10 Academic (MPM2D) Unit 3: Algebra & Quadratic Models Solving Quadratic Equations by Factoring Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved.

Section 2.1 Arithmetic Sequences and Series

Homework Questions.

Grade 11 Functions (MCR3U) Unit 4: Trigonometric Functions Graphing Trig. functions (Amplitude and Period) Mr. Choi © 2018 E. Choi – MCR3U - All Rights.

Grade 11 Functions (MCR3U)

11.6 Binomial Theorem & Binomial Expansion

Grade 12 Advanced Functions (MHF4U) Unit 3: Trigonometry Addition and Subtraction Formulas Mr. Choi © 2017 E. Choi – MHF4U - All Rights Reserved.

Grade 10 Academic (MPM2D) Unit 3: Algebra & Quadratic Models Common Factors & Factoring by Grouping Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved.

Grade 10 Academic (MPM2D) Unit 3: Algebra & Quadratic Models Factoring Difference of Squares Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved.

Grade 10 Academic (MPM2D) Unit 5: Trigonometry Geometry RevieW

Grade 12 Advanced Functions (MHF4U) Unit 3: Trigonometry Double Angle Formulas Mr. Choi © 2017 E. Choi – MHF4U - All Rights Reserved.

Grade 10 Academic (MPM2D) Unit 3: Algebra & Quadratic Models Factoring Simple Trinomials Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved.

Geometric Sequences and series

Section 2 – Geometric Sequences and Series

Unit 5 Polynomial Operations

Geometric Sequences and Series

12.2 – Geometric Sequences and Series

Grade 11 University: (MCR3U) Unit 3: Exponential Functions Solving Exponential Equations 1 Mr. Choi © 2018 E. Choi – MCR3U - All Rights Reserved.

Grade 11 University: (MCR3U) Unit 1: Algebra & Quadratic Functions Review of Factoring 2 Mr. Choi © 2017 E. Choi – MCR3U - All Rights Reserved.

Grade 11 University: (MCR3U) Unit 1: Algebra & Quadratic Functions Intersection of Linear and Non-linear Relations Mr. Choi © 2017 E. Choi – MCR3U - All.

Warm Up Write the first 4 terms of each sequence:

Grade 11 Functions (MCR3U) Unit 4: Trigonometry Trigonometric Identities 2 Mr. Choi © 2018 E. Choi – MCR3U - All Rights Reserved.

Grade 11 Functions (MCR3U) Unit 4: Trigonometry Trigonometric Identities 1 Mr. Choi © 2018 E. Choi – MCR3U - All Rights Reserved.

Section 12.3 Geometric Sequences; Geometric Series

Presentation transcript:

Grade 11 Functions (MCR3U) Unit 5: Pascal’s Triangle, Binomial Theorem, Sequences & Series Geometric Series Mr. Choi © 2018 E. Choi – MCR3U - All Rights Reserved

Recall: What is a Series? A series is the sum of terms of a sequence. The sum of the first n terms in a sequence is denoted as Sn. S5  the sum of the first 5 terms Geometric Series © 2018 E. Choi – MCR3U - All Rights Reserved

Geometric Series Recall: A sequence like 2, 6, 18, 54,…, is called a Geometric sequence with a = 2 and r = 3. Therefore a series that corresponds to this geometric sequence is 2 + 6 + 18 + 54 +…, is called a geometric series. Geometric Series © 2018 E. Choi – MCR3U - All Rights Reserved

Geometric Series Formula Recall: Geometric Sequence formula The Sum of the First n terms of a Geometric Series Multiply this series by r Subtract this two rows All the terms except the first and last are cancelled out or Formula 2 (Optional) Formula 1 Geometric Series © 2018 E. Choi – MCR3U - All Rights Reserved

Example 1 – Sum of a Series given the first term Find the sum of the first 9 terms of the geometric series. Geometric Series © 2018 E. Choi – MCR3U - All Rights Reserved

Example 2 – Sum of a Series given the first term Find the sum of the first 14 terms of the geometric series. Geometric Series © 2018 E. Choi – MCR3U - All Rights Reserved

Example 3 – Sum of a series given first and last terms Find the sum of the geometric series 5 + 20 + 80 + ... + 20480. 7 7 There are 7 terms in the series. Geometric Series © 2018 E. Choi – MCR3U - All Rights Reserved

Example 4 – Word Problem A tennis tournament has 128 entrants. A player is dropped from the competition after losing one match. Winning players go on to another match. What is the total number of matches that will be played in this tournament? Since there are 2 players/match, the first term Half of the players drop out because they lost. 7 Since there will be a single match played at the end of the tournament, the last term is 1. 7 Therefore a total of 127 matches will be played in this tournament. Geometric Series © 2018 E. Choi – MCR3U - All Rights Reserved

Homework: Text Book: P.408 #1-7 (ac for all), 9-12 Work Sheet: Check the website for updates Geometric Series © 2018 E. Choi – MCR3U - All Rights Reserved

End of Lesson Geometric Series © 2018 E. Choi – MCR3U - All Rights Reserved