Chapter 5: Sequeces ad Series 1. Sequeces 2. Arithmetic ad Geometric Sequeces 3. Summatio Notatio 4. Arithmetic Series 5. Geometric Series 6. Mortgage Paymets
LESSON 1 SEQUENCES I Commo Core Algebra I, you studied sequeces, which are ordered lists of umbers. Sequeces are extremely importat i mathematics, both theoretical ad applied. A sequece is formally defied as a fuctio that has as its domai the set of positive itegers, i.e. 1, 2, 3,...,. Exercise #1: A sequece is defied by the equatio a 2 1 (a) Fid the first three terms of this sequece, deoted by a1, a2, ad a 3.. (b) Which term has a value of 53? (c) Explai why there will ot be a term that has a value of 70. Recall that sequeces ca also be described by usig recursive defiitios. recursively, terms are foud by operatios o previous terms. Whe a sequece is defied Exercise #2: A sequece is defied by the recursive formula: f f 1 5 with (a) Geerate the first five terms of this sequece. Label each term with proper fuctio otatio. f 1 2. (b) Determie the value of f 20. Hit thik about how may times you have added 5 to 2.
Exercise #3: Determie a recursive defiitio, i terms of f iclude a startig value. 5, 10, 20, 40, 80, 160,, for the sequece show below. Be sure to t 1 2 Exercise #4: For the recursively defied sequece 2 (1) 18 (3) 456 (2) 38 (4) 1446 t ad t 1 2, the value of t 4 is Exercise #5: Oe of the most well-kow sequeces is the Fiboacci, which is defied recursively usig two previous terms. Its defiitio is give below. 1 2 ad f f f f f 1 1 ad 2 1 Geerate values for f 3, f 4, f 5, ad f 6 (i other words, the ext four terms of this sequece).
It is ofte possible to fid algebraic formulas for simple sequece, ad this skill should be practiced. Exercise #6: Fid a algebraic formula Recall that the domai that you map from will be the set a, similar to that i Exercise #1, for each of the followig sequeces. 1, 2, 3,...,. (a) 4, 5, 6, 7,... (b) 2, 4, 8,16,... (c) 5 5 5 5,,,,... 2 3 4 1 1 1 (d) 1, 1, 1, 1,... (e) 10, 15, 20, 25, (f) 1,,,,... 4 9 16 Exercise #7: Which of the followig would represet the graph of the sequece a 2 1? Explai your choice. (1) y (2) y (3) y (4) y Explaatio:
LESSON 1 HOMEWORK SEQUENCES FLUENCY 1. Give each of the followig sequeces defied by formulas, determie ad label the first four terms. A variety of differet otatios is used below for practice purposes. (a) f 7 2 (b) a 2 5 (c) t 2 (d) 3 t 1 1 2. Sequeces below are defied recursively. Determie ad label the ext three terms of the sequece. (a) f 1 4 ad f f 1 8 (b) a a 1 a 1 ad 1 24 2 (c) b b 1 2 with 1 5 2 b (d) f 2f 1 ad f 1 4
3. Give the sequece 7, 11, 15, 19,..., which of the followig represets a formula that will geerate it? (1) a 4 7 (3) a 3 7 (2) a 3 4 (4) a 4 3 4. A recursive sequece is defied by a 1 2 a a 1 with a1 0 ad a2 1. Which of the followig represets the value of a 5? (1) 8 (3) 3 (2) 7 (4) 4 5. Which of the followig formulas would represet the sequece 10, 20, 40, 80, 160, a (3) a (1) 10 (2) 52 a 10 2 (4) a 2 10
6. For each of the followig sequeces, determie a algebraic formula, similar to Exercise #4 from the lesso, that defies the sequece. Do ot write it usig the recursive formula. (a) 5, 10, 15, 20, (b) 3, 9, 27, 81, (c) 1 2 3 4,,,,... 2 3 4 5 7. For each of the followig sequeces, state a recursive defiitio. Be sure to iclude a startig value or values. (a) 8, 6, 4, 2, (b) 2, 6, 18, 54, (c) 2, 2, 2, 2,... APPLICATIONS 8. A tilig patter is created from a sigle square ad the expaded as show. If the umber of squares i each patter defies a sequece, the determie the umber of squares i the seveth patter. Explai how you arrived at your choice. Ca you write a recursive defiitio for the patter?
REASONING 9. Cosider a sequece defied similarly to the Fiboacci, but with a slight twist: 1 2 with f f f f f 1 2 ad 2 5 Geerate terms f(3), f(4), f(5), f(6), f(7), f(8), ad f(9). The, determie the value of f 25.
LESSON 2 ARITHMETIC AND GEOMETRIC SEQUENCES I Commo Core Algebra I, you studied two particular sequeces kow as arithmetic (based o costat additio to get the ext term) ad geometric (based o costat multiplyig to get the ext term). I this lesso, we will review the basics of these two sequeces. ARITHMETIC SEQUENCE RECURSIVE DEFINITION Give f 1, the 1 f f d or give a1 the a a 1 d where d is called the commo differece ad ca be positive or egative. Exercise #1: Geerate the ext three terms of the give arithmetic sequeces. (a) f f 1 6 with 1 3 f 1 2 (b) a a 1 ad a1 2 2 Exercise #2: For some umber t, the first three terms of a arithmetic sequece are 2t, 5t 1, ad 6t 2. (a) What is the value of t? (b)what is the umerical value of the fourth term?
It is importat to be able to determie a geeral term of a arithmetic sequece based o the value of the idex variable (the subscript). The ext exercise walks you through the thikig process ivolved. Exercise #3: Cosider a a 1 3 with a1 5. (a) Determie the value of a2, a3, ad a 4. (b) How may times was 3 added to 5 i order to produce a 4? (c) Use your result from part (b) to quickly fid the value of a 50. th (d) Write a formula for the term of a arithmetic sequece, a, based o the first term, a 1, d ad. Exercise #4: Give that a1 6 ad a4 18 are members of a arithmetic sequece, determie the value of a 20.
Exercise #5: I a arithmetic sequece t = t 1 5. If t 1 = 3 determie the values of t 6 ad t 25. Show the calculatios that lead to your aswers. Exercise #6: Fid the umber of terms i a arithmetic sequece whose first two terms term are -3 ad 4 ad whose last term is 116. Geometric sequeces are defied very similarly to arithmetic, but with a multiplicative costat istead of a additive oe. GEOMETRIC SEQUENCE RECURSIVE DEFINITION Give 1 the 1 f f f r or give a 1, the a a 1 r where r is called the commo ratio ad ca be positive or egative ad is ofte fractioal. Exercise #7: Geerate the ext three terms of the geometric sequeces give below. (b) f f 1 1 with (a) a1 4 ad r 2 3 f 1 9 (c) t t 1 2 with t1 3 2
Ad, like arithmetic, we also eed to be able to determie ay give term of a geometric sequece based o the first value, the commo ratio, ad the idex. Exercise #8: Cosider a1 2 ad a a 1 3. (a) Geerate the value of a 4. (b) How may times did you eed to multiply 2 by 3 i order to fid a 4. (c) Determie the value of a 10. th (d) Write a formula for the term of a geometric sequece, a, based o the first term, a 1, r ad. Exercise #9: Give a 1 = -2 ad a 2 = 8 are the first two terms of a geometric sequece, determie the values of a 4 ad a 8. Show calculatios that lead to your aswer.
LESSON 2 HOMEWORK ARITHMETIC AND GEOMETRIC SEQUENCES FLUENCY 1. Geerate the ext three terms of each arithmetic sequece show below. (b) f f f (a) a1 2 ad d 4 1 8 with 1 10 (c) a1 3, a2 1 2. I a arithmetic sequece t t 1 7. If t 1 5 determie the values of t4 ad t 20. Show the calculatios that lead to your aswers. 3. If x 4, 2x 5, ad 4x 3 represet the first three terms of a arithmetic sequece, the fid the value of x. What is the fourth term? 4. If f 1 12 ad f f 1 4 the which of the followig represets the value of 40 (1) 148 (3) 144 (2) 140 (4) 172 f?
5. I a arithmetic sequece of umbers a1 4 ad a 6 46. Which of the followig is the value of a 12? (1) 120 (3) 92 (2) 146 (4) 106 6. The first term of a arithmetic sequece whose commo differece is 7 ad whose 22 d term is give by a22 143 is which of the followig? (1) 25 (3) 7 (2) 4 (4) 28 7. Geerate the ext three terms of each geometric sequece defied below. (a) a1 8 with r 1 (b) a 3 a 1 2 ad 1 16 a (c) f f f 1 2 ad 1 5 8. Give that a1 5 ad a2 15 are the first two terms of a geometric sequece, determie the values of a ad a. Show the calculatios that lead to your aswers. 3 10
9. I a geometric sequece, it is kow that a1 1 ad a4 64. The value of a 10 is (1) 65, 536 (3) 512 (2) 262,144 (4) 4096 APPLICATIONS 10. The Koch Sowflake is a mathematical shape kow as a fractal that has may fasciatig properties. It is created by repeatedly formig equilateral triagles off of the sides of other equilateral triagles. Its first six iteratios are show to the right. The perimeters of each of the figures form a geometric sequece. (a) If the perimeter of the first sowflake (the equilateral triagle) is 3, what is the perimeter of the secod sow flake? Note: the dashed lies i the secod sowflake are ot to be couted towards the perimeter. They are oly there to show how the sowflake was costructed. (b) Give that the perimeters form a geometric sequece, what is the perimeter of the sixth sowflake? Express your aswer to the earest teth. (c) If this process was allowed to cotiue forever, explai why the perimeter would become ifiitely large.
LESSON 3 SUMMATION NOTATION Much of our work i this uit will cocer addig the terms of a sequece. I order to specify this additio or summarize it, we itroduce a ew otatio, kow as summatio or sigma otatio that will represet these sums. This otatio will also be used later i the course whe we wat to write formulas used i statistics. SUMMATION (SIGMA) NOTATION ia f a f a 1 f a 2 f f i where i is called the idex variable, which starts at a value of a, eds at a value of, ad moves by uit icremets (icrease by 1 each time). Exercise #1: Evaluate each of the followig sums. (a) 5 2i (b) i3 3 2 k (c) k 1 2 j2 2 j 5 (d) 1 i (e) 2k 1 (f) ii1 i1 2 k 0 3 i1
4 1 Exercise #2: Which of the followig represets the value of i? (1) 1 10 (3) 25 12 i1 (2) 9 4 (4) 31 24 Exercise #3: Cosider the sequece defied recursively by a a 1 2 a2 ad a1 0 ad a2 1. Fid the value of 7 i4 a i It is also good to be able to place sums ito sigma otatio. These aswers, though, will ot be uique. Exercise #4: Express each sum usig sigma otatio. Use i as your idex variable. First, cosider ay patters you otice amogst the terms ivolved i the sum. The, work to put these patters ito a formula ad sum. (a) 916 25 100 (b) 6 3 03 15 (c) 1 1 1 5 625 25 5
Exercise #5: Which of the followig represets the sum 3612 24 48? (1) 5 3 i (3) i1 4 i0 1 6 i 4 (2) 3 2 i (4) i0 48 i3 i Exercise #6: Some sums are more iterestig tha others. Determie the value of reasoig. This is kow as a telescopig series (or sum). 99 1 1. Show your 1 i1 i i
LESSON 3 HOMEWORK SUMMATION NOTATION FLUENCY 1. Evaluate each of the followig. Place ay o-iteger aswer i simplest ratioal form. (a) 5 3 2 4i (b) k 1 i2 k 0 (c) 2j 1 0 j 2 (d) 3 2 i 2 2 1 (e) 1 k i1 k 0 3 (f) log10 i i1 (g) 4 (h) 1 1 4 i2 4 i2 2 i 1 i 2 1 (i) 3 k 0 1 2 256 k
2. Which of the followig is the value of 4k 1 (1) 53 (3) 37 (2) 45 (4) 80 4? k 0 3. The sum 7 7 2 i is equal to i4 (1) 15 8 (3) 3 4 (2) 3 2 (4) 7 8 4. Write each of the followig sums usig sigma otatio. Use k as your idex variable. Note, there are may correct ways to write each sum (ad eve more icorrect ways). (a) 2 4 8 64 128 (b) 1 1 1 1 1 (c) 4914 44 49 1 4 9 81 100 5. Which of the followig represets the sum 2 510 82 101? 6 10 2 (3) j 1 j1 j1 (1) 4j 3 103 11 (4) 4 j 1 (2) j 2 j3 j0
6. A sequece is defied recursively by the formula b 4b 1 2 b 2 with b1 1 ad b2 3. What is the value of 5 bi? Show the work that leads to your aswer. i3 REASONING 7. A curious patter occurs whe we look at the behavior of the sum 2k 1 (a) Fid the value of this sum for a variety of values of below: 2. k 1 k 4 : 2k 1 2 : 2 1 k 1 4 k 1 3 k 5 : 2k 1 3: 2 1 k 1 5 k 1 (b) What types of umbers are you summig? What types of umbers are the sums? 2 1 196. (c) Fid the value of such that k k 1
LESSON 4 ARITHMETIC SERIES A series is simply the sum of the terms of a sequece. The fudametal defiitio/otio of a series is below. THE DEFINITION OF A SERIES If the set a, a, a,... represet the elemets of a sequece the the series, S, is defied by: 1 2 3 S ai i1 I truth, you have already worked extesively with series i previous lessos almost aytime you evaluated a summatio problem. Exercise #1: Give the arithmetic sequece defied by a1 2 ad a a 1 5, the which of the followig is the value of S 5 5 a? i1 i (1) 32 (3) 25 (2) 40 (4) 27 The sums associated with arithmetic sequeces, kow as arithmetic series, have iterestig properties, may applicatios ad values that ca be predicted with what is commoly kow as raibow additio. Exercise #2: Cosider the arithmetic sequece defied by a1 3 ad a a 1 2. The series, based o the first eight terms of this sequece, is show below. Terms have bee paired off as show. (a) What does each of the paired off sums equal? (b) Why does it make sese that this sum is costat? 35 7 9111315 17 (c) How may of these pairs are there? (d) Usig your aswers to (a) ad (c) fid the value of the sum usig a multiplicative process. (e) Geeralize this ow ad create a formula for a arithmetic series sum based oly o its first term, a 1, its last term, a, ad the umber of terms,.
Exercise #3: Which of the followig is the sum of the first 100 atural umbers? Show the process that leads to your choice. (1) 5,000 (3) 10,000 (2) 5,100 (4) 5,050 SUM OF AN ARITHMETIC SERIES Give a arithmetic series with terms, a1, a2,..., a, the its sum is give by: S a1 a 2 Exercise #4: Fid the sum of each arithmetic series described or show below. (a) The sum of the sixtee terms give by: 10 6 2 46 50. (b) The first term is 8, the commo differece, d, is 6 ad there are 20 terms (c) The last term is a12 29 ad the commo differece, d, is 3. (d) The sum 5811 77.
Exercise #5: The first ad last terms of a arithmetic series are 7 ad -121, respectively, ad the series has a sum of -1026. How may terms are i this series? Exercise #6: Kirk has set up a college savigs accout for his so, Maxwell. If Kirk deposits $100 per moth i a accout, icreasig the amout he deposits by $10 per moth each moth, the how much will be i the accout after 10 years?
LESSON 4 HOMEWORK ARITHMETIC SERIES FLUENCY 1. Which of the followig represets the sum of 310 87 94 if the arithmetic series has 14 terms? (1) 1,358 (3) 679 (2) 658 (4) 1,276 2. The sum of the first 50 atural umbers is (1) 1,275 (3) 1,250 (2) 1,875 (4) 950 3. If the first ad last terms of a arithmetic series are 5 ad 27, respectively, ad the series has a sum 192, the the umber of terms i the series is (1) 18 (3) 14 (2) 11 (4) 12
4. Fid the sum of each arithmetic series described or show below. (a) The sum of the first 100 eve, atural umbers. (b) The sum of multiples of five from 10 to 75, iclusive. (c) A series whose first two terms are 12 ad 8, respectively, ad whose last term is 124. (d) A series of 20 terms whose last term is equal to 97 ad whose commo differece is five. 5. For a arithmetic series that sums to 1,485, it is kow that the first term equals 6 ad the last term equals 93. Algebraically determie the umber of terms summed i this series.
APPLICATIONS 6. Arligto High School recetly istalled a ew black-box theatre for local productios. They oly had room for 14 rows of seats, where the umber of seats i each row costitutes a arithmetic sequece startig with eight seats ad icreasig by two seats per row thereafter. How may seats are i the ew black-box theatre? Show the calculatios that lead to your aswer. 7. Simeo starts a retiremet accout where he will place $50 ito the accout o the first moth ad icrease his deposit by $5 per moth each moth after. If he saves this way for the ext 20 years, how much will the accout cotai i pricipal?
8. The distace a object falls per secod while oly uder the ifluece of gravity forms a arithmetic sequece with it fallig 16 feet i the first secod, 48 feet i the secod, 80 feet i the third, etcetera. What is the total distace a object will fall i 10 secods? Show the work that leads to your aswer. 9. A large gradfather clock strikes its bell oce at 1:00, twice at 2:00, three times at 3:00, etcetera. What is the total umber of times the bell will be struck i a day? Use a arithmetic series to help solve the problem ad show how you arrived at your aswer.
LESSON 5 GEOMETRIC SERIES Just as we ca sum the terms of a arithmetic sequece to geerate a arithmetic series, we ca also sum the terms of a geometric sequece to geerate a geometric series. Exercise #1: Give a geometric series defied by the recursive formula a1 3 ad a a 1 2, which of the followig is the value of S 5 5 a? i1 (1) 106 (3) 93 (2) 75 (4) 35 i SUM OF A FINITE GEOMETRIC SERIES For a geometric series defied by its first term, a 1, ad its commo ratio, r, the sum of terms is give by: Exercise #2: Which of the followig represets the sum of a geometric series with 8 terms whose first term is 3 ad whose commo ratio is 4? (1) 32,756 (3) 42,560 (2) 28,765 (4) 65,535 S a 1 1 r 1 r a a r 1 r 1 1 or S
Exercise #3: Fid the value of the geometric series show below. Show the calculatios that lead to your fial aswer. 612 24 768 Exercise #4: Maria places $500 at the begiig of each year ito a accout that ears 5% iterest compouded aually. Maria would like to determie how much moey is i her accout after she has made her $500 deposit at the ed of 10 years. At, that a give $500 has grow to t-years after it was placed ito this accout. (a) Determie a formula for the amout, (b) At the ed of 10 years, which will be worth more: the $500 ivested i the first year or the fourth year? Explai by showig how much each is worth at the begiig of the 11th year. (c) Based o (b), write a geometric sum represetig the amout of moey i Maria s accout after 10 years. (d) Evaluate the sum i (c) usig the formula above.
Exercise #5: A perso places 1 pey i a piggy bak o the first day of the moth, 2 peies o the secod day, 4 peies o the third, ad so o. Will this perso be a millioaire at the ed of a 31 day moth? Show the calculatios that lead to your aswer. Exercise #6: Fid the sum of the first 10 terms of the sequece 1 5, 2 5, 4 5, 8 5, Exercise #7: A geometric series has a first term of 8 ad last term of 1/16. Its commo ratio is ½. Fid the value of this series.
LESSON 5 HOMEWORK GEOMETRIC SERIES FLUENCY 1. Fid the sums of geometric series with the followig properties: (a) a1 6, r 3 ad 8 (b) a 1 1 20, r, ad 6 (c) 2 a1 5, r 2, ad 10 128 2. If the geometric series 54 36 has seve terms i its sum the the value of the sum is 27 (1) 4118 27 (2) 1274 3 (3) 1370 9 (4) 8241 54 3. A geometric series has a first term of 32 ad a fial term of series is (1) 19.75 (3) 22.5 (2) 16.25 (4) 21.25 1 ad a commo ratio of 4 1. The value of this 2 4. Which of the followig represets the value of has i it. 8 i0 i 3 256? Thik carefully about how may terms this series 2 (1) 19,171 (3) 22,341 (2) 12,610 (4) 8,956
5. A geometric series whose first term is 3 ad whose commo ratio is 4 sums to 4095. The umber of terms i this sum is (1) 8 (3) 6 (2) 5 (4) 4 6. Fid the sum of the geometric series show below. Show the work that leads to your aswer. 1 27 9 3 729 APPLICATIONS 7. I the picture show at the right, the outer most square has a area of 16 square iches. All other squares are costructed by coectig the midpoits of the sides of the square it is iscribed withi. Fid the sum of the areas of all of the squares show. First, cosider the how the area of each square relates to the larger square that surrouds (circumscribes) it. 8. A college savigs accout is costructed so that $1000 is placed the accout o Jauary 1 st of each year with a guarateed 3% yearly retur i iterest, applied at the ed of each year to the balace i the accout. If this is repeatedly doe, how much moey is i the accout after the $1000 is deposited at the begiig of the 19 th year? Show the sum that leads to your aswer as well as relevat calculatios.
9. A ball is dropped from 16 feet above a hard surface. After each time it hits the surface, it rebouds to a height that is 3 of its previous maximum height. What is the total vertical distace, to the earest foot, the ball has 4 traveled whe it strikes the groud for the 10 th time? Write out the first five terms of this sum to help visualize.
LESSON 6 MORTGAGE PAYMENTS Mortgages are ot just made o houses. They are large amouts of moey borrowed from a bak o which iterest is calculated. The iterest is calculated o a regular basis (usually mothly). Regular paymets are made o the amout of moey owed so that over time the pricipal (origial amout borrowed) is paid off as well as ay iterest o the amout owed. This is a complex process that ivolved geometric series. Formula: m = r P( 12 ) 1 (1+ r 12 ) m = mortgage paymet (mothly) p = pricipal (loa amout) r = iterest rate (decimal) = umber of paymets Examples: 1.) Tom s house will cost $300,000. He eeds to pay a deposit of 10% ad will pay the remaiig 90% over 30 years at 8% iterest rate per year. How much does Tom eed per moth to buy this house? 2.) Calculate the mothly paymet eeded to pay off a $200,000 loa at 4% yearly iterest rate over a 20 year period. Now calculate the pay off period to be 30 years. How much less is the mothly paymet?
3.) You would like to buy a home priced at $200,000. You pla to make a paymet of 10% of the purchase price. (a) Compute the total mothly paymet for a 30-year mortgage at 4.8% aual iterest. (b) What is the total iterest paid over the life of the loa? (c) Compute the total mothly paymet ad the total iterest paid over the life of a 20-year mortgage at 4.8% aual iterest. (d) Why would someoe choose a 20-year mortgage over a 30-year mortgage? Why might aother perso choose a 30-year mortgage?
4.) Suppose you would like to buy a home priced at $180,000. You qualify for a 30-year mortgage at 4.5% aual iterest. (a) Calculate the total mothly paymet ad the iterest paid over the life of the loa if you make a 3% dow paymet. (b) Calculate the total mothly paymet ad the iterest paid over the life of the loa if you make a 10% dow paymet. (c) Calculate the total mothly paymet ad the iterest paid over the life of the loa if you make a 20% dow paymet.
5.) The followig amortizatio table shows the amout of paymets to pricipal ad iterest o a $100,000 mortgage at the begiig ad ed of a 30-year loa. (a) Describe the chages i the amout of pricipal paid each moth as the ed of the loa approaches. (b) Describe the chages i the amout of iterest paid each moth as the ed of the loa approaches.
LESSON 6 HOMEWORK MORTGAGE PAYMENTS Complete the followig questios. Show all work, icludig formulas. 1.) Christopher wats to buy a $200,000 home with a 30-year mortgage at 4.5% aual iterest payig 10% dow. (a) What is the mothly paymet o the house? (b) The graph below depicts the amout of your paymet from part (b) that goes to the iterest o the loa ad the amout that goes to the pricipal o the loa. Explai how you ca tell which graph is which. 2.) I the summer of 2014, the average listig price for homes for sale i the Hollywood Hills was $2, 663, 995. Suppose you wat to buy a home at that price with a 30-year mortgage at 5. 25% aual iterest, payig 10% as a dow paymet. What is your total mothly paymet o this house?
3.) Suppose that you wat to buy a house that costs $175,000. You ca make a 5% dow paymet, ad a. Fid the mothly paymet for a 30-year mortgage o this house at a 4.25% iterest rate. b. Fid the mothly paymet for a 15-year mortgage o this house at the same iterest rate. 4.) Suppose that you would like to buy a home priced at $450, 000. You qualify for a 30-year mortgage at 4. 5% aual iterest a. Calculate the total mothly paymet ad the total iterest paid over the life of the loa if you make a 3% dow paymet. b. Calculate the total mothly paymet ad the total iterest paid over the life of the loa if you make a 10% dow paymet. c. Calculate the total mothly paymet ad the total iterest paid over the life of the loa if you make a 20% dow paymet.