MCR3U Unit 7 Lesson Notes

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1 7.1 Arithmetic Sequences Sequence: An ordered list of numbers identified by a pattern or rule that may stop at some number or continue indefinitely. Ex. 1, 2, 4, 8,... Ex. 3, 7, 11, 15 Term (of a sequence): A single value or object in a sequence. Each number in a sequence is called a term. Each term is identified by its position in the sequence An arithmetic sequence is a sequence in which the difference between consecutive terms is a constant. The difference between consecutive terms of an arithmetic sequence is called a common difference. Ex. 13, 11, 9, 7,... In this example the common difference is -2 The following sequence is arithmetic, ie. the terms increase by the same amount. Figure 1 Figure 2 Figure 3 Figure 4... Figure n Recursive Formula: or General Term of an Arithmetic Sequence: A recursion formula is a formula by which each term of a sequence is generated from the preceding term or terms. t 1 = a, t n = t n-1 + d,n > 1 t n = a + (n-1)d A recursion formula has 2 parts: First part: begins the sequence ex. t 1 = 1 Second part: used to write the terms, one after the other. ex. t n = t n-1 + 2, n > 1 A recursion formula allows you to calculate any term provided you know the preceding term. Here we can calculate any term (nth term) in a sequence providing we know the first term ('a') and the common difference ('d'). How many blocks would be needed for the 47 th term? If you had 135 blocks, what figure number could you make? 1

2 Ex1. Determine if the sequence is arithmetic. If it is, state the general term, and the recursive formula. a) 35, 33, 31, 29,... b) c) 2

3 Ex2. Determine the first 3 terms. If arithmetic, state the common difference. a) t n = 3n b) f(n) = 10 - n Ex3. Determine the number of terms in the following arithmetic sequence. 0.6, 0.72, 0.84,..., Assignment: p. 424 #5-11, 13,15,16 (parts acde for all) 3

4 7.2 Geometric Sequences Geometric Sequence: a sequence that has a common ratio between consecutive terms. The common ratio "r" can be found by taking any term and dividing it by the term before. 3, 12, 48, 192,... Recursive Formula: General Term of a Geometric Sequence: t 1 = a, t n = rt n-1,n > 1 t n = ar n-1 Ex1. A company has 3 kg, of radioactive material that must be stored. After one year, 95% of the radioactive material remains. How much radioactive material will be left at the end of 10 years? Ex2. Determine if the following sequences are arithmetic, geometric or neither. If they are arithmetic or geometric state the general term for the sequence. a) 31, 32, 34, 37,... b) 29, 19, 9, -1,... c) 128, 96, 72, 54,... 4

5 Ex3. Given a geometric sequence with t 7 = -192 and t 9 = -768, determine t 16 Ex4. Determine the general term and the recursive formula for the following geometric sequence. 1, 0.2, 0.04, Assignment: p. 430 #3, 5-12 (parts acde) 13, 14ab 5

6 7.3/7.4 Exploring Recursive Sequences and Creating Rules to Define Sequences A recursion formula is a formula by which each term of a sequence is generated from the preceding term or terms. A recursion formula is a recipe for generating the terms, starting with the first term A recursion formula allows you to calculate any term provided you know the preceding term Examples: 1. Write the first 4 terms for the following sequences given their recursion formula. a) b) c) d) 6

7 The Fibonacci sequence is the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34,... Each number after the first two numbers, 1 and 1, is the sum of the preceding two numbers a. Write a recursion formula for the Fibonnaci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34,... b. Determine the 10th, 11th, and 12th, term. Write a recursion formula for the following sequences. a) 2, -6, 18, -54,... b) 4, 5, 7, 10, 14,... 7

A sequence has a general term if an algebraic rule using the term number, n,

8 A sequence is an ordered list of numbers that may or may not follow a predictable pattern. A sequence has a general term if an algebraic rule using the term number, n, can be found to generate each term. Determine the next 3 terms and state the general term of the sequence. a) b) c) p. 439 #1, 2, 3ab, 5, 7, 11 8

7.5 Arithmetic Series Arithmetic Series: The sum of the terms of an arithmetic sequence.

com/algebra/19 sequences series/06 gauss problem arithmetic series 01 What Gauss actually did was find a pattern by reversing the

..+ 101 + 101 + 101 This represents twice the sum of the series that we want, therefore: or We can use this concept to develop a

The sum of a general arithmetic series: S = a + (a+d) + (a+2d) +.

9 7.5 Arithmetic Series Arithmetic Series: The sum of the terms of an arithmetic sequence. For the story of Carl Friedrich Gauss and how it relates to the sum of an arithmetic series. sequences series/06 gauss problem arithmetic series 01 What Gauss actually did was find a pattern by reversing the series and Adding vertically: This represents twice the sum of the series that we want, therefore: or We can use this concept to develop a general formula for the sum of an arithmetic series with first term "a" and a common difference "d". The sum of a general arithmetic series: S = a + (a+d) + (a+2d) [a+(n 3)d] + [a+(n 2)d] + [a+(n 1)d] S = [a+(n 1)d] + [a+(n 2)d] + [a+(n 3)d] (a+2d) + (a+d) + a 2S = [2a + (n 1)d] + [2a + (n 1)d] + [2a + (n 1)d] [2a + (n 1)d] + [2a + (n 1)d] + [2a + (n 1)d] 2S = n x [2a + (n 1)d] Therefore the formula for the sum of a general arithmetic series is: 9

10 Ex.1: Determine the sum of the first 30 terms of Ex.2: Determine the sum of:

11 Ex.3: Determine the sum of an arithmetic series in which t 5 = 21 and the last term t 36 = 72 Ex.4: In an arithmetic series S 1 = 1, S 2 = 3, S 3 = 6. Determine S 12 and t 100 Assignment 7.5: p. 452 #4 7, 10 12, 15,16 11

12 7.6 Geometric Series Recall a Geometric Sequence: 2, 10, 50, 250, 1250, 6250,... with a = 2, r = 5 and whose general term is: t n = ar n 1 Geometric Series is just the sum of a geometric sequence. Therefore the geometric series for the above geometric sequence is: with a = 2, r = 5 and whose general term is: t n = ar n 1 What if we wanted the sum of the first 6 terms of the series above: S = Trick: multiply through by "5" 5S = Subtract: S 5S = Therefore: 4S = S = / 4 S = 7812 Now let's find the Sum for a general geometric series: S = a + ar + ar 2 + ar ar n 2 + ar n 1 multiply through by "r" rs = ar + ar 2 + ar ar n 2 + ar n 1 + ar n Subtract: S rs = a ar n Factor out the S: S(1 r) = a ar n or S = a ar n or S n = a(1 r n ) or S n = a(r n 1) 1 r 1 r r 1 12

13 Ex.1: Calculate t 8 and S 8 for each of the following geometric series: a) b)

14 c) 5 10x + 20x 2... Ex.2: Find the sum of: Assignment 7.6: p. 459 #3 8,11 14

7.7 PASCAL'S TRIANGLE AND THE BINOMIAL THEOREM Pascal's Triangle is a triangular arrangement of number with a 1 in the first row, and 1 and 1 in the second row.

15 7.7 PASCAL'S TRIANGLE AND THE BINOMIAL THEOREM Pascal's Triangle is a triangular arrangement of number with a 1 in the first row, and 1 and 1 in the second row. Each number in the succeeding rows is the sum of the two numbers above it in the preceding row Position of Terms: A term in Pascal's Triangle can be represented by t n,r where n is the horizontal row number and r is the diagonal row number. Ex. Expand the following * Compare the n th power to the n th row's elements in Pascal's Triangle. Now compare these to the coefficients of the expanded polynomial. Look for other patterns in the expanded polynomials. * Powers of binomials can be expanded using patterns. The coefficients on the expansion of (a + b) n can be found in row n of Pascal's Triangle. 15

16 Ex. Expand the following using the Binomial Theorem and Pascal's Triangle a. b. c. d. Assignment 7.7 : p. 466 #1, 2ac, 4aef, 5ace 16

Examples of Finite Sequences (finite terms) Examples of Infinite Sequences (infinite terms)

Examples of Finite Sequences (finite terms) Examples of Infinite Sequences (infinite terms) Math 120 Intermediate Algebra Sec 10.1: Sequences Defn A sequence is a function whose domain is the set of positive integers. The formula for the nth term of a sequence is called the general term. Examples