Objectives 1. Understand and use terminology and notation involved in sequences. A number sequence is any list of numbers with a recognizable pattern.

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1 1. Understand and use terminology and notation involved in sequences A number sequence is any list of numbers with a recognizable pattern. The members of a sequence are called terms (or sometimes members). The terms are usually given a single name like u or x or t with a subscript that indicates the position in the sequence. The subscript is called the index. Sometimes the index starts at 0, other times at 1. If it's not stated, we usually assume that the index starts at 1. We generally use the letter n to represent a general term. (Note: n sometimes also refers to the total number of terms in a sequence. Watch the context.) u n is the n th term in a sequence of u values. The notation {u n } represents a function that generates the sequence defined by using u n as the n th term. Example: {½n + 4} generates the sequence 4.5, 5, 5.5, 6, Note how this differs from a continuous function: 6A: #1,2cfi,3,4def (Number patterns) Sequence Continuous {u n } = {½n + 4} f(x) = ½x + 4 Domain: n an integer > 0 Domain: x a real number 6A: #1,2cfi,3,4def (Number patterns) 11/27 6A: Number sequences 1

2 Go over Exp/Log and Transformations Quizzes 6A: #1,2cfi,3,4def (Number patterns) Discuss questions only 1. Recognize and use various representations of a sequence. 2. Understand sequences as discrete data. 6B: #1,2,3gh,4,5 (General Term & Notation) There are many ways to represent a sequence: List terms 3, 6, 9, 12,... Use words Starting with 0 and 1 the next term is the sum of the previous two Use a formula u n = 3 + 4n Use pictures Note that the dots are not connected with a line since a sequence is discrete data. The mathematical formula is referred to as the general term because it can be used to find the value of any term in the sequence specifically, the n th term. General terms are represented with a symbol and a subscript. For example, u 1 or T 7 Pay attention to the fact that there are two ideas being represented: The symbol represents the value of the term. The subscript represents the position of the term in the sequence. So T 7 is the value of the seventh term! Other notation and notes: The domain of n (the indices) is n Z +. That is, n = 1, 2, 3,... (sometimes finite) {u n } represents the sequence of values generated using u n as the general term. For example, {3n 5} represents 2, 1, 4, 7,... 6B: #1,2,3gh,4,5 (General Term & Notation) 11/29 6B: General Term 2

3 6B: #1,2,3gh,4,5 (General Term & Notation) Discuss questions only 1. Understand and use ideas underlying arithmetic sequences 2. Know how to calculate the n th term given u 1 and d, or two terms 9,13,17,21,... Find the pattern. 6C.1: #2,3,5,7def,8cd,9a,11 (Arithmetic sequences) 6C.2: #2,3 (Arithmetic sequence problems) QB: #3,6 (IB Arithmetic Sequences) An Arithmetic Sequence is a sequence in which each term differs from the previous one by the same amount. Algebraically: {u n } is arithmetic if and only if u n+1 = u n + d for all integers n > 0 The difference between each term, d, is called the common difference. Let's develop some features of arithmetic sequences: Three terms in an arithmetic sequence are 5, x, 9. What is the common difference? What is the middle term, x? What if the three terms were 200, x, 400? 123, x, 725 In general, what is the middle term of three terms a, x, b of an arithmetic sequence? In an arithmetic sequence the middle term of three successive terms is the arithmetic mean of the first and third. Hence the name! Consider the arithmetic sequence 14, 17, 20, What is the value of the next term? What is the value of the 8 th term? What is the value of the 21 st term? What is the value of the n th term? What is the value of the n th term of an arithmetic sequence with first term u 1 and common difference d?... + d + d + d + d u 1, u 2, u 3, u 4,... u n 1, u n, u n+1, u n+2... u 1 +1d u 1 +2d u 1 +3d u 1 +(n 1)d The general expression for the n th term of an arithmetic sequence with first term u 1 and common difference d is: u n = u 1 + (n 1)d Examples of problems involving arithmetic sequences: In an arithmetic sequence, the first term is 5 and the 4 th term is 40. Find the 2 nd term. 6C.1: #2,3,5,7def,8cd,9a,11 (Arithmetic sequences) 6C.2: #2,3 (Arithmetic sequence problems) QB: #3,6 (IB Arithmetic Sequences) 11/29 6C: Arithmetic sequences 3

4 6C.1: #2,3,5,7def,8cd,9a,11 (Arithmetic sequences) Present 2c,3 (show that),5,7f,8d,9a,11 6C.2: #2,3 (Arithmetic sequence problems) Present 2 QB: #3,6 (IB Arithmetic Sequences) Present 3,6.1 &.2 6D.1: #1c,2b,3d,7,9bd,10a (Geometric Sequences) 6D.2: #2,4 (Geometric Sequence problems) 1. Understand and use ideas underlying geometric sequences 2. Know how to calculate the n th term given u 1 and r, or two terms 4,12,36,108,... Find the pattern. A Geometric Sequence is a sequence in which the ratio of successive terms is constant. Algebraically: The ratio of successive terms, r, is called the common ratio. Let's develop some features of geometric sequences: Three terms in an geometric sequence are 3, x, 12. What is the common ratio? What is the middle term, x? What if the three terms were 3, x, 15? In general, what is the middle term of three terms a, x, b of an geometric sequence? In a geometric sequence the middle term of three successive terms is the geometric mean of the first and third. Hence the name! Consider a geometric sequence with a first term of 3 and a common ratio of Write out the first five terms without performing any multiplications 2. Graph these values using n as the independent variable 3. Have you seen this pattern before? 4. Rewrite the first five terms without performing any multiplications 5. See if you see a pattern that enables you to write the general term n. 6. Generalize using u, r, and n. n Calculation u n 1 u 1 = Initial value u 1 2 u 2 = u 1 r u 2 = u 1 r 3 u 3 = u 2 r = (u 1 r) r u 3 = u 1 r 2 4 u 4 = u 3 r = ((u 1 r) r) r u 4 = u 1 r n u n = u n 1 r = (((u 1 r) r)...) r u n = u 1 r (n 1) r r r... r u 1, u 2, u 3, u 4,... u n 1, u n, u n+1, u n+2... u 1 r 1 u 1 r 2 u 1 r 3 u 1 r n 1 The general expression for the n th term of a geometric sequence with first term u 1 and common ratio r is: u n = u 1 r n 1 Use logs! On a calculator you can generate a sequence very easily: 2nd LIST/OPS/seq (5) Expr: the general term Variable: which letter in the general term should be treated as "n"? Start: the starting index (often = 1) End: the ending index (calculator can't go on forever you know!) Step: how much to increment the variable each time (usually 1) Notice that you can enter a sequence as a list definition! Try the previous table on your calculator Remember that geometric progressions are exponential and thus can model population and other kinds of exponential growth and decay. An example: Desert Academy's current enrollment is 200 (more or less). Suppose that it increases by 4% per year. What would the enrollment be after 5 years? In what year would we exceed our maximum capacity of 250? Very similar to interest: Value after n years = u 1 (1 + r) n = 200 (1.04) 5 = students What about the second question? Can you write the equation we are trying to solve? (1.04) n Yes, you know logs! Finish it: years Another example a.i) 138 a.ii) 381 b) 35 th week 12/3 6D: Geometric sequences 4

5 6D.1: #1c,2b,3d,7,9bd,10a (Geometric Sequences) Present 7,10a 6D.2: #2,4 (Geometric Sequence problems) Present Understand compound interest as a geometric sequence 6D.3: #3,6,9,10 (Compound Interest) QB: #9a c,10a c.i,12a (IB Geometric Sequences) Applications of geometric sequences: Compound Interest Imagine that you have $100 in a bank and that they pay you 3% per year. How much do you have in the bank after n years? To understand this more easily, notice that adding 3% to something is the same as multiplying it by a +.03a = a(1 +.03) = a(1.03) = 1.03a So let's make a table of values: n Year Calculation Balance at end of year 1 0 $ (1.03)(100) $ (1.03)(1.03)(100) $ (1.03)(1.03)(1.03)(100) $ (1.03)(1.03)(1.03)(1.03)(100) $ n + 1 n (1.03) n (100) Compound Interest (annual compounding) If you invest u 1 dollars for n years at an interest rate of r % per year the value after n years is given by Alternatively, u n = u 1 (1 + r % ) n 1 u n+1 = u 1 (1 + r % ) n In the second formulation, u n+1 is the amount after n periods. Note: The book uses a slightly different formula. Some formulas define r as the interest rate. Others define r as a "growth multiplier" or a "growth rate" in which the "1 +" is already included. Pay attention! Either way, it's a geometric sequence. Periodic Compounding: If interest is compounded over a different time period, you need to adjust r and n. Understand that r is the interest rate earned each period. So if the annual rate is 5% and the compounding is every month, the rate per period is 5%/12 or % (= ). Each year represents 12 periods. So in this case, if you started with P dollars after n years you would have: Annual rate divided by months each year. 12n is the number of periods. Your calculator is a friend indeed: APPS/FINANCE/TVM Solver... # of years of investment interest rate per year "Present value" of your money (<0 because you invest it) "Payment" (=0 use this for finding loan payments) "Future value" of your money (at the end of the time period) Payments/year (=1 use this if you're calculating loan balance) "Compoundings/year" Payment timing (When, in each period, you make the payment) Enter the values you know Move cursor to value you want Press SOLVE (ALPHA/ENTER) 6D.3: #3,6,9,10 (Compound Interest) QB: #9a c,10a c.i,12a (IB Geometric Sequences) 12/4 6D.3 Compound Interest 5

6 Last day to make up MB missing assignments is Thu 12/13 (7 days from now). 6D.3: #3,6,9,10 (Compound Interest) Present 6,10 (See TVM Solver) 1. Understand and use sigma notation 6E: #1bcf,2cf,3,4c,5 (Sigma Notation) A series is a sum of the terms in a sequence. For example, if we have a sequence {u n } = 1,2,3,4,5,...,n we can create series that represent the sum of part (or all!) of the sequence. For example, S 5 = = 15. S 5 is called the 5 th partial sum of the sequence and S n is the n th partial sum. It would be convenient to develop some notation to describe sums like this, especially if there are a lot of terms in the sum. Any thoughts? What if the terms are more complicated as in the sequence {3n + 2 n + 1 }? Will your notation work? Sigma Notation: We use the Greek letter sigma to represent sums as follows: n = the index of the last term in the series = an expression defining the terms being added i = the letter indicates which letter in the expression is the index 1 = the first number to use for the index in the series Note: We use S n to represent series (that is, sums) and u i to represent the terms of a sequence. Write out a few: Properties of sigma notation sums can be broken apart constants, c, can be factored out summing a constant is like multiplying Some examples And don't forget your calculator: LIST/MATH/sum( (5) Can define a sequence in the sum! Try one of the above Last day to make up MB missing assignments is Thu 12/13 (7 days from now). Be sure to do QB solidly! 6E: #1bcf,2cf,3,4c,5 (Sigma Notation) QB: #9a c,10a c.i,12a (IB Geometric Sequences) 12/6 6E Series Intro 6

7 Last day to make up MB missing assignments is Thu 12/13 (3 days from now). 6E: #1bcf,2cf,3,4c,5 (Sigma Notation) Present #5 QB: #9a c,10a c.i,12a (IB Geometric Sequences) Questions?... the sum of an arithmetic sequence 1. Understand and solve problems involving arithmetic series 6F: #1 11odd (Arithmetic Series) QB: #1,2,4,5 (IB Arithmetic Series) Consider the arithmetic series S n = How many terms are in it? or solve 534 = (n 1) Can we generalize? How many terms in an arithmetic series that starts with u 1, has a common difference of d and ends with u n? Add one for the Number of terms in an arithmetic sequence fencepost at the end. given the first, last, and common difference What is the sum of the terms? If you start with the original... S n = rewrite it in the opposite order S n = Add them to get twice the sum. 2S n = We know the number of terms n = (534 12)/3 + 1 (last term is a fencepost!) So we can get use a product: 2S n = = And the final result is half that: S n = ( )/2 = 95550/2 = n = # of terms divide by 2 sum of first and last Properties of Arithmetic Series Given and arithmetic sequence with first term u 1 and common difference, d: The sum of the first n terms is given by: Since the n th term is given by u n = u 1 + d(n 1) an equivalent form is: The sum of an arithmetic series is The 12 th term is 147 and the 26th term is 49. Find n. Without a calculator, find = 40/2( ) = = 3500 Not for the faint of heart: Can you derive the above formula using sigma notation? Last day to make up MB missing assignments is Thu 12/13 (3 days from now). 6F: #1 11odd (Arithmetic Series) QB: #1,2,4,5 (IB Arithmetic Series) Bell, E. T Men of Mathematics. New York: Simon and Schuster. (See chapter 14, "The Prince of Mathematicians: Gauss," pp ) Shortly after his seventh birthday Gauss entered his first school, a squalid relic of the Middle Ages run by a virile brute, one Büttner, whose idea of teaching the hundred or so boys in his charge was to thrash them into such a state of terrified stupidity that they forgot their own names. More of the good old days for which sentimental reactionaries long. It was in this hell hole that Gauss found his fortune. Nothing extraordinary happened during the first two years. Then, in his tenth year, Gauss was admitted to the class in arithmetic. As it was the beginning class none of the boys had ever heard of an arithmetic progression. It was easy then for the heroic Büttner to give out a long problem in addition whose answer he could find by a formula in a few seconds. The problem was of the following sort, , where the step from one number to the next is the same all along (here 198), and a given number of terms (here 100) are to be added. It was the custom of the school for the boy who first got the answer to lay his slate on the table; the next laid his slate on top of the first, and so on. Büttner had barely finished stating the problem when Gauss flung his slate on the table: "There it lies," he said "Ligget se'" in his peasant dialect. Then, for the ensuing hour, while the other boys toiled, he sat with his hands folded, favored now and then by a sarcastic glance from Büttner, who imagined the youngest pupil in the class was just another blockhead. At the end of the period Büttner looked over the slates. On Gauss' slate there appeared but a single number. To the end of his days Gauss loved to tell how the one number he had written down was the correct answer and how all the others were wrong. Gauss had not been shown the trick for doing such problems rapidly. It is very ordinary once it is known, but for a boy of ten to find it instantaneously by himself is not so ordinary. 12/10 6F Arithmetic series 7

8 Last day to make up MB missing assignments is Thu 12/13 (2 days from now). 6F: #1 11odd (Arithmetic Series) Present #7,11 QB: #1,2,4,5 (IB Arithmetic Series) Present all...the sum of a geometric sequence 1. Understand and solve problems involving geometric series (n 1) How many terms in 3, 6, 12, ,608? u n = u 1 r What is the sum of the terms? 6G.1: #2cd,3,4bc,5,6 (Finite Geometric Series) 6G.2: #1,2,4b,5b,6,8 (Infinite Geometric Series) QB: #7 12 (IB Geometric Series) (n 1) 196,608 = 3 2 (n 1) = 2 log 2 (65536) = n 1 16 = n 1 n = 17 Start with a general series Multiply both sides by r (increases each power of r by 1) Notice that S n = u 1 + u 1 r + u 1 r u 1 r n 2 + u 1 r n 1 rs n = u + u 1 r n 1 r + u 1 r u 1 r n 2 + u 1 r n 1 u 1 r + u 1 r u 1 r n 2 + u 1 r n 1 = S n u 1 Condense RHS A little algebra... will take us to... the result we want: Properties of Geometric Series Given a geometric sequence with first term u 1 and common ratio r, the sum of n terms is given by: Recall that the n th term is given by: (n 1) u n = u 1 r Can you write the above as an equation using sigma notation instead of S n? Think about what this formula will do as the number of terms increases. If r > 1, each successive term gets bigger and the sum will diverge (see numerator). But if r < 1, each successive term gets smaller. What happens as n? The series will converge to Sum of an Infinite Geometric Series The limiting sum as n of an infinite series with r < 1 is given by: What happens when r < 0. Hmmmm... These are called alternating series. Last day to make up MB missing assignments is Thu 12/13 (2 days from now). 6G.1: #2cd,3,4bc,5,6 (Finite Geometric Series) 6G.2: #1,2,4b,5b,6,8 (Infinite Geometric Series) QB: #7 12 (IB Geometric Series) 12/11 6G Geometric series 8

9 Last day to make up MB missing assignments is Today! 6G.1: #2cd,3,4bc,5,6 (Finite Geometric Series) 6G.2: #1,2,4b,5b,6,8 (Infinite Geometric Series) QB: #7 12 (IB Geometric Series) Present as needed, take questions About 40% of your final exam will involve sequences and series and transformations. The rest will be based on ideas from earlier in the course. You may use only the official SL formula sheet as support. The final will have some problems on which you may use a calculator and others where you may not. Suggested review: > Review 1B: #1 11 odd (Quadratics) > Review 2B: #7 9 (Functions) > Review 3A: #7 11 odd (Exponentials) > Review 4A: #5 12 (Logarithms) > Review 5C: #5,6 (Transformations) > Review 6C: #2 10 even (Sequences & Series) > QB problems and tests Solving quadratic equations Completing the square Factoring Quadratic formula Using the discriminant to solve problems Graphing and using graphs of quadratic functions Solving Evaluating functions Domain & Range Composing functions Notation (g o f)(x) Inverse functions Rational functions Graphs Asymptotes Zeroes Laws of exponents Graphs of exponential functions Solving exponential equations Growth & Decay Laws of logarithms Graphs of logarithmic functions Solving logarithmic equations Growth & Decay Vertical and horizontal Shifts, stretches & compressions Reflections around axes General sequences Arithmetic and Geometric sequences Finding number of terms Finding n th term General series Sigma notation Arithmetic and Geometric series Finding sum of terms Infinite geometric series 12/11 & 12/12 Review 9