Fractals and Sequences PDF

The diagrams shown here are the first four iterations of a fractal called the Koch snowflake. What do you notice about: how each pattern is created from the previous one? the perimeter as you move from the first iteration through the fourth iteration? How is it changing? the area en...

The diagrams shown here are the first four iterations of a fractal called the Koch snowflake. What do you notice about: how each pattern is created from the previous one? the perimeter as you move from the first iteration through the fourth iteration? How is it changing? the area enclosed as you move from the first iteration to the fourth iteration? How is it changing? What changes would you expect in the fifth iteration? How would you measure the perimeter at the fifth iteration if the original triangle had sides of 1 m in length? If this process continues forever, how can an infinite perimeter enclose a finite area? Developing inquiry skills Does mathematics always reflect reality? Are fractals such as the Koch snowflake invented or discovered? Think about the questions in this opening problem and answer any you can. As you work through the chapter, you will gain mathematical knowledge and skills that will help you to answer them all. Before you start Click here for help with this skills check You should know how to: Skills check 1 Solve linear equations: 1 Solve each equation: eg 2(x- 2) - 3(3*+ 7) = 4 a 2x + 10 = -4x - 8 2x - 4 - 9x - 2 \ =4 b 3a - 2(2a + 5) = -12 - 7x - 25 = 4 c (x + 2)(x- 1) = (x+ 5)(x- 2) -lx-29 _ -29 7 2 Perform operations (+ - x +) with fractions 2 Simplify: and simplify using order of operations: Cc | \J\ p -+ U irs | t"- | oo P -s I N| 03 i* n + X I | r> For example, 5, 10, 20, 40,... or 200, 100, 50, 25,... Example 1 Find an expression tor the general term for each of the following sequences and state whether they are arithmetic, geometric or neither. — a 3,6,9,12,... b 3,-12,48,-192,... C 2,10,30,230,... TD I | | | ^ IT, r- O' a The sequence can be written as This sequence is the positive multiples of 3. 3 x 1, 3 x 2, 3 x 3,... The general term is u = 3n. This is an arithmetic sequence because 3 is being This is an arithmetic sequence. added to each term. b The sequence can be written as This is a geometric sequence because each term is 3 x 1, 3 x -4, 3 x 16, 3 x -64 multiplied by -4. which can be expressed as 3 x (-4)°, 3 x (-4)1, 3 x (-4)2, 3 x (-4)3,... The general term is h m= 3(-4)"-' This is a geometric sequence. c The sequence can be written as This is a geometric sequence because each term is 2 x 1, 2 x 5, 2 x 25, 2 x 125,... or multiplied by 5. 2 x 5°, 2 x 51, 2 x 52, 2 x 53,... The general term is uit = 2 x 5M_1 This is a geometric sequence. Continued on next page o i 7  FROM PATTERNS TO GENERALIZATIONS: SEQUENCES AND SERIES d The sequence can be written as ------, Consider the numerators and denominators ,. 3 2+3 as two separate sequences. or 4+3 6+3 1 2 The numerators are the positive integers. 3 + (2x0) 3 + (2x 1) 3 + (2 x 2) 3 + (2x3) The denominators, 3, 5, 7, 9,... , are the The general term is odd positive integers with h - 3 and 2 K= being added to each term. ” 3 + 2{n - l) n 3 + 2/7 - 2 n 2/7 + 1 The sequence is neither arithmetic nor Each sequence has a first term, a second geometric. term, a third term, etc, so n will always be a positive integer. 1 Write down the next three terms in each 3 For each of the following real-life situations, sequence: give the general term and state whether a -8, -11, -14, -17,... b 9, 16, 25, 36,... the sequence is arithmetic, geometric or neither. c 6, 12, 18, 24,... d 1000, 500, 250, 125,... a Ahmad deposits $100 in a savings account —| every month. After the first month, the «N | fA fA - + | a | > 31 | | Tt* balance is $ 100; after the second month, C o 1=1 language? For an infinite series, the upper limit is °°. An example of an infinite X geometric series is ]T3x2,r. M= 1 Example 4 For each of the following finite series in sigma notation, find the terms and calculate the sum: a b c 'Yjln-n2 nisi n=i m =1 a (-1)'(1 )2 + (-1 )2(2)2 + (-1 )3(3)2 + (-1 )4(4)2 Substitute n = 1 to find the first term, = -l +4-9+16 n = 2 for the second term and so on. = 10 The last term is found when you substitute the upper limit of n = 4. Remember to add up all the terms once you have found them. b 1000 3-1 Using the GDC: n = 5.92... When using a graph or solver on the GDC, be sure to give the decimal answer first, The minimum number of terms is 6. then give the final answer. or 3(3" -l) —------ - > 1000 3-1 Using a table on the GDC: when n = 5, Sn =363 When using a table on the GDC, be sure to give both crossover values, then give the when n = 6, S„ = 1092 final answer. The minimum number of terms is 6. asm 1 In an arithmetic sequence, the b i Find u 28* first term is -8 and the sum of ii Hence, find S28* the first 20 terms is 790. Find how many terms it takes a Find the common difference. for the sum to exceed 2000. 34  1.3 2 In an arithmetic series, call five other employees, and so r SAr. = 1900 and uACi = 106. Find on. How many rounds of phone N um ber and algebra 40 40 the value of the first term and calls are needed to reach all 2375 the common difference. employees? 3 The sum of an infinite geometric 8 A geometric sequence has all series is 20, and the common positive terms. The sum of the ratio is 0.2. Find the first term of first two terms is 15 and the sum this series. to infinity is 27. 4 The sum of an infinite geometric a Find the value of the common series is three times the first term. ratio. Find the common ratio of this b Hence, find the first term. series. 9 The first three terms of an 5 Create two different infinite infinite geometric sequence are geometric series that each have a m - I, 6, m + 8. sum of 8. a Write down two expressions for r. 6 In a geometric sequence, the b i Find two possible values of m. fourth term is 8 times the first ii Hence, find two possible term. The sum of the first 10 values of r. TOK terms is 2557.5. Find the 10th c i Only one of these r term of this sequence. How do values forms a geometric ? A large company created a phone sequence where an infinite mathematicians tree to contact all employees in sum can be found. Justify reconcile the case of an emergency. Each of your choice for r. fact that some the five vice presidents calls five ii Hence, calculate the sum to conclusions employees, who in turn each infinity. conflict with intuition? Developing inquiry skills Returning to the chapter opening investigation about the Koch snowflake, the enclosed area can be found using the sum of an infinite series. In the second iteration, since the sides of the new triangles are 1 the length of the sides of the original triangle, their areas must be j -j =- of its area. If the area of the original triangle is 1 square unit, then the total area of the three new triangles is 3| -J. i Find the total area for the third and fourth iterations. ii How can you use what you have learned in this section to find the total area of the Koch snowflake? iii How does the area of a Koch snowflake relate to the area of the initial triangle? 35  FROM PATTERNS TO GENERALIZATIONS: SEQUENCES AND SERIES 1.4 Applications of arithmetic and geometric patterns As you have seen in previous sections, arithmetic and geometric sequences and series can be applied to many real-life situations. In this section, you will explore two of these applications: interest and population growth. Interest is the charge for borrowing money. Interest can also be used to describe money earned on an investment account. You will focus on two types of interest, simple and compound. Simple interest is interest paid on the initial amount borrowed, saved or invested (called the principal] only and not on past interest. Compound interest is paid on the principal and the accumulated interest. Interest paid on interest! Investigation 8 Part 1: An amount of $1000 is invested into an account that pays 5% per annum simple interest on a monthly basis. 1 Calculate the amount of interest paid per month. 2 Find the value of this investment after the first three months. 3 What do you notice about this sequence? What sort of sequence is it? 4 Write the general formula that can be used to find the amount of the investment after n months. 5 Use your formula to calculate the value of investment after two years. Part 2: An amount of $1000 is invested into an account that pays 5% per annum compound interest. Interest is compounded monthly. 1 Calculate the amount of interest paid after the first month. 2 Find the value of the investment after the first month. 3 Continue this process for the second and third months. 4 What do you notice about this sequence? What sort of sequence is it? 5 Write the general formula that can be used to find the amount of the investment after n months. 6 Use your formula to calculate the value of investment after two years. ? Compare your answers from the final question of each part. Which investment is worth more; the simple interest or the compound interest? o 36  1.4 8 H.I.UJ.HIMI Which type of series models simple interest? Which type N um ber and algebra models compound interest? 9 MW How are outcomes and growth patterns related? From the previous investigation, you should have concluded that simple interest can be modelled by an arithmetic series, and compound interest can be modelled by a geometric series. \ For simple interest, assuming A represents the accumulated amount, P is the principal, r is the annual rate and n is the time in years: Initial amount: A=P After the first year: A = P + Pr = P( 1 + r) After the second year: A = P( 1 + r) -F Pr = P( 1 + r + r) = P( 1 + 2r) After the third year: A = P{ 1 + 2r) + Pr = P( 1 + 2r + r) = P( 1 + 3r) After the /7th year: A = P( 1 + nr) For compound interest, making the same assumptions as above: Initial amount: A=P After the first year: A = P + Pr = P( 1 +r) After the second year: A = P( 1 + r) + P( 1 + r)r = P( 1 +r)( 1 + r) = P(1 + r)2 After the third year: A = P( 1 + r)2 + P( 1 + r)2r = P( \ +r)2(l +r) = P(l +r)3 After the /7 th year: A = P(1 + r)n Interest can be paid over any time period—for example, yearly, monthly, quarterly, and so on. To calculate compound interest over a non-year period, we use this formula: ( \nt A = P\ i + I I , where A is the final amount (principal + interest], P is the principal, r is the annual interest rate expressed as a decimal, n is the number of compoundings in a year, and t is the total number of years. 3?  FROM PATTERNS TO GENERALIZATIONS: SEQUENCES AND SERIES Example 22 Sebastian took out a loan for a new car that cost $35 000. The bank offered him 2.5% per annum simple interest for five years. Calculate the total value Sebastian has to repay the bank. Example 23 Habib put $5000 into a savings account that pays 4% interest per annum, compounded monthly. How much will be in the account after four years if Habib does not deposit or withdraw any money? Since the interest is paid monthly. 0.04 r =--------- and 12 nt = 4x 12 = 48 48 A = 500of 1 + ° 041 V 12 J A = $5865.99 Example 24 Leslie got a student loan of $12 090 CAD to pay her tuition at the University of Toronto this year. If Leslie has to repay $16000 CAD in two years, what is the annual interest rale if the interest on the loan is compounded monthly? 16 000 = 12 090^1 + “ j —KF 1+ — = 23j~ (3x2x1) r = "" «C5_5!(8-5)! 81 8! 8 5 ! 3! 8 x 7 x 6 x 5 x 4 x -3-x-2-*+ *C,i ” (5x 4 x 3x 2x 1)(4*3*4-) £ 8x7x 6x5x4 * 5“ 5x4> EXAM HINT We need proofs in mathematics to show that the mathematics we use every QED is an abbreviation day is correct, logical and sound. for the Latin phrase There are many different types of proofs (deductive, inductive, by “quod erat demon contradiction), but we will use algebraic proofs in this section. strandum” which means The goal of an algebraic proof is to transform one side of the mathematical “what was to be statement until it looks exactly like the other side. demonstrated" or “what was to be shown”. The One rule is that you cannot move terms from one side to the other. Imagine sign with three bars (=) that there is an imaginary fence between the two sides of the statement and is the identity symbol, terms cannot jump the fence! which means that the At the end of a proof, we write a concluding statement, such as two sides are equal by LHS = RHS or QED. definition. L________ _____________ 2 55  FROM PATTERNS TO GENERALIZATIONS: SEQUENCES AND SERIES Example 35 Prove that - = — + 8 4 2 LHS RHS 'C —| < First, simplify the RHS using a common denominator. + I | O ^ N< O —| N| + rf '"T | rj* Once simplified completely, in order for the RHS to match 2 the LHS, multiply by — (which is equivalent to 1). Don't forget a concluding statement at the end. Prove that -2(a - 4) + 3(2a + 6) 4 a Prove that - 6(a - 5) = -2(a - 28). x-2^_____ ___ 3* - 6 _ x + \ What is the role of the Prove that (x - 3)2 + 5: 6x x X2 + x 3 mathematical community + 14. For what values of x in determining the validity does this mathematical of a mathematical proof? Prove that — = —-— statement not hold true? m m +1 m +m Chapter summary A sequence (also called a progression) is a list of numbers written in a particular order. Each number in a sequence is called a term. A finite sequence has a fixed number of terms. An infinite sequence has an infinite number of terms. A formula or expression that mathematically describes the pattern of the sequence can be found for the general term, Ufi. A sequence is called arithmetic when the same value is added to each term to get the next term. A sequence is called geometric when each term is multiplied by the same value to get the next term. A recursive sequence uses the previous term or terms to find the next term. The general term will include the notation Ufj_ {, which means “the previous term”. A series is created when the terms of a sequence are added together. A sequence or series can be either finite or infinite. A finite series has a fixed number of terms. For example, P + 5 + 3 + l-l—1H—3 is finite because it ends after the sixth term. An infinite series continues indefinitely. For example, the series 10-I-8-I-6H-4-F... is infinite because the ellipsis indicates that the series continues indefinitely. * 56  1 The formula for any term in an arithmetic sequence is: un = w, + (n -1 )d Number and algebra The formula for any term in a geometric sequence is: un = uxrn~x. The sum, Sfj, of an arithmetic series can be expressed as:Sn = — (w, + ut1). The sum, Sfj, of an arithmetic series can also be expressed as: Stl = \[2w,+ («-!)*- 1) “(1~r") r*l 1 -r The sum of a converging infinite geometric series is: 5 = It]., | r\ < 1 00 l - r Simple interest is interest paid on the initial amount borrowed, saved or invested (called the principal] only and not on past interest. Compound interest is paid on the principal and the accumulated interest. Interest paid on interest! n\ We can calculate combinations using the formula nC = --------rr 6 r r!(fl-r)! A mathematical proof is a series of logical steps that show one side of a mathematical statement is equivalent to the other side for all values of the variable. We need proofs in mathematics to show that the mathematics we use every day is correct, logical and sound. The goal of an algebraic proof is to transform one side of the mathematical statement until it looks exactly like the other side. One rule is that you cannot move terms from one side to the other. At the end of a proof, we write a concluding statement, such as LHS = RHS or QED. Developing inquiry skills Return to the opening problem. How has your understanding of the Koch snowflake changed as you have worked through this chapter? What features of, for example, the ninth iteration can you now work out from what you have learned? 5?  FROM PATTERNS TO GENERALIZATIONS: SEQUENCES AND SERIES r--------------------- Chapter review V_____ Click here for a mixed review exercise _____ 1 For each of the following sequences, d How much water has been drained a Identify whether the sequence is after 15 minutes? arithmetic, geometric or neither, e How long will it take to drain the tank? b If it is arithmetic or geometric, find an expression for un> Marie-Jeanne is experimenting c If it is arithmetic or geometric, find the with weights of indicated term. lOOg attached to a d If it is arithmetic or geometric, find the spring. indicated sum. i 3, 6, 18,... wg, S12 She records the weight and length ii -16, -14, -12,... w10, S8 of the spring after iii 2000, 1000, 500,... w9, 5? attaching the iv X3x 2"-,»5,S10 weight. v The consecutive multiples of 5 greater Number of 100 g weights 1 2 3 4 than 104, ur S9 attached to the spring 2 In an arithmetic sequence, u6 = -5 and Length of spring (cm) 45 49 53 5? u9 = -20. Find S2Q. 3 Write down the first five terms for the recur a Write a general formula that represents sive sequence un = -2un ] + 3 with u{ = -4. the length of the spring if n weights are attached. 4 For the geometric series 0.5 - 0.1 + 0.02... b Calculate the supposed length of a spring Sn = 0.416. Find the number of terms in the if a 1 kg weight is attached, series. c Explain any limitations of Marie- 5 For a geometric progression, = 4.5 and u7 Jeanne's experiment. = 22.78125. Find the value of the common d If the spring stretches to 101 cm, what is ratio and the first term. the total weight that was attached to the 6 Which of the following sequences has an spring? infinite sum? Justify your choice and find Kostas gets a four-year bank loan to buy a that sum. new car that is priced at €20 987. After the A-,--,—,... B 0.06,0.12,0.24,... four years, Kostas will have paid the bank a 4 8 16 total of €22 960. What annual interest rate ? How many terms are in the sequence did the bank give him if the interest was 4, 7, 10,..., 61? compounded monthly? 8 In a geometric sequence, the fourth term is The first seven numbers in row 14 of Pascal's 8 times the first term. If the sum of the first triangle are 1, 13, 78, 286, 715, 1287, 1716. 10 terms is 765, find the 10th term of the a Complete the row and explain your sequence. strategy. 9 Three consecutive terms of a geometric b Explain how you can use your answer sequence are x - 3, 6 and x + 2. Find all in part a to find the 15th row. State the possible values of x. terms in the 15th row. 10 A tank contains 55 litres of water. Water 14 Using the binomial theorem, expand flows out at a rate of 7% per minute, (3x-y)6. a Write a sequence that represents the ^ volume left in the tank after 1 minute, Find the coefficient of the term in x2 in 2 minutes, 3 minutes, etc. the expansion of — - 4x b What kind of sequence did you write? Justify your answer. IB In the binomial expansion of — - 5xs c How much water is left in the tank after 10 minutes? the sixth term contains x25. Solve for n. 58  1 1? a Find the term in x5 in the expansion of b Calculate, to the nearest year, how (x-3)9. long Brad must wait for the value of Number and algebra b Hence, find the term in x6 in the the investment to reach $12 000. expansion of -2x(x - 3)9. (5 marks) ( ^’ P2: Find the coefficient of the term in x5 in 18 In the expansion of — + — I , the constant the binomial expansion of (3 +x)(4 + 2x)s. 3 x, (4 marks) is i 12 640 pjn(j value of k. 27 2 6 PI: The coefficient of x2 in the binomial KT7| expansion of (l + 3x)" is 495. 19 Prove that: Determine the value of n. (6 marks) (2x- 1 )(x — 3) -3(x-4)2 = -x2- 31* + 51. 2 ? P2: Find the constant term in the expansion 20 a Prove that ^-l£z6.4zi6=^lZ£±12 x+4 x‘+2 x of \x'-- (4 marks) x b For what values of x does this mathe 2 8 P2: a Find the binomial expansion of matical statement not hold true? 1 Exam-style questions H in ascending powers of x. (3 marks) Hence, or otherwise, find the term PI: a Find the binomial expansion of independent of x in the binomial -—I in ascending powers of x. ^/ (3 marks) expansion of b Using the first three terms from (4 marks) the above expansion, find an approximation for 0.975s. (3 marks) 2 9 P2: A convergent geometric series has sum to infinity of 120. PI: The 15th term of an arithmetic series is Find the 6th term in the series, given 143 and the 31st term is 183. that the common ratio is 0.2. (5 marks) a Find the first term and the common 3 0 PI: The second term in a geometric series | difference. (5 marks) 20 b Find the 100th term of the series. is 180 and the sixth term is —. 9 (2 marks) Find the sum to infinity of the series. P2: Angelina deposits $3000 in a savings (7 marks) account on 1 January 2019, earning n-l “> 3 1 P2: Find the value of £ (1.6" - \2n +1), giving compound interest of 1.5% per year, flaO a Calculate how much interest (to your answer correct to 1 decimal place. the nearest dollar) Angelina would (6 marks) earn after 10 years if she leaves the 3 2 PI: A ball is dropped from a vertical money alone. (3 marks) height of 20 m. b In addition to the $3000 deposited Following each bounce, it rebounds to a on January 1st 2019, Angelina deposits a further amount of $1200 vertical height of 2 its previous height. 6 into the same account on an annual Assuming that the ball continues to basis, beginning on 1st January 2020. bounce indefinitely, show that the Calculate the total amount of money maximum distance it can travel is in her account at the start of January 220m. (5 marks) 2030 (before she has deposited her 3 3 PI: Prove the binomial coefficient identity money for that year). (4 marks) n -1 n-\\ P2: Brad deposits $5500 in a savings (6 marks) account which earns 2.75% compound k-1 interest per year. 34 P2: Find the sum of all integers between 500 and 1400 (inclusive) that are not a Determine how much Brad's inves tment will be worth after 4 years. divisible by 7. (7 marks) (3 marks) 59