UNIT #5 SEQUENCES AND SERIES Lesso # Sequeces Lesso # Arithmetic ad Geometric Sequeces Lesso #3 Summatio Notatio Lesso #4 Arithmetic Series Lesso #5 Geometric Series Lesso #6 Mortgage Paymets COMMON CORE ALGEBRA II - UNIT #5
COMMON CORE ALGEBRA II - UNIT #5
Name: Date: SEQUENCES COMMON CORE ALGEBRA II 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 the set of positive itegers, i.e.,, 3,...,. Exercise #: A sequece is defied by the equatio a (a) Fid the first three terms of this sequece, deoted by a, a, 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. Whe a sequece is defied recursively, terms are foud by operatios o previous terms. Exercise #: A sequece is defied by the recursive formula: f f (a) Geerate the first five terms of this sequece. Label each term with proper fuctio otatio. 5 with f. (b) Determie the value of f 0. Hit thik about how may times you have added 5 to. Exercise #3: Determie a recursive defiitio, i terms of f iclude a startig value. 5, 0, 0, 40, 80, 60,, for the sequece show below. Be sure to t Exercise #4: For the recursively defied sequece () 8 (3) 456 () 38 (4) 446 t ad t, the value of t 4 is COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #
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. ad f f f f f ad 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. a, similar to that i Exercise #, for each of the followig Exercise #6: Fid a algebraic formula sequeces. Recall that the domai that you map from will be the set,, 3,...,. (a) 4, 5, 6, 7,... (b), 4, 8,6,... (c) 5 5 5 5,,,,... 3 4 (d),,,,... (e) 0, 5, 0, 5, (f),,,,... 4 9 6 Exercise #7: Which of the followig would represet the graph of the sequece a? Explai your choice. y y y () () (3) (4) y Explaatio: COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #
Name: Date: SEQUENCES COMMON CORE ALGEBRA II HOMEWORK FLUENCY. 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 (b) a 5 (c) t (d) 3 t. Sequeces below are defied recursively. Determie ad label the ext three terms of the sequece. (a) f 4 ad f f 8 (b) a a a ad 4 (c) b b with 5 f f ad f 4 b (d) 3. Give the sequece 7,, 5, 9,..., which of the followig represets a formula that will geerate it? () a4 7 (3) a3 7 () a3 4 (4) a4 3 4. A recursive sequece is defied by a a a with a 0 ad a. Which of the followig represets the value of a 5? () 8 (3) 3 () 7 (4) 4 5. Which of the followig formulas would represet the sequece 0, 0, 40, 80, 60, a (3) a () 0 () 5 a 0 (4) a 0 COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #
6. For each of the followig sequeces, determie a algebraic formula, similar to Exercise #4, that defies the sequece. (a) 5, 0, 5, 0, (b) 3, 9, 7, 8, (c) 3 4,,,,... 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,, (b), 6, 8, 54, (c),,,,... 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: with f f f f f ad 5 Geerate terms f 3, f 4, f 5, f 6, f 7, ad f 8. The, determie the value of 5 f. COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #
Name: Date: ARITHMETIC AND GEOMETRIC SEQUENCES COMMON CORE ALGEBRA II 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. Give ARITHMETIC SEQUENCE RECURSIVE DEFINITION f, the f f d or give a the a a d where d is called the commo differece ad ca be positive or egative. Exercise #: Geerate the ext three terms of the give arithmetic sequeces. 3 6 with f (b) a a ad a (a) f f Exercise #: For some umber t, the first three terms of a arithmetic sequece are t, 5t, ad 6t. What is the umerical value of the fourth term? Hit: first set up a equatio that will solve for t. 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 3 with a 5. (a) Determie the value of a, 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, d ad. COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #
Exercise #4: Give that a 6 ad a4 8 are members of a arithmetic sequece, determie the value of a 0. Geometric sequeces are defied very similarly to arithmetic, but with a multiplicative costat istead of a additive oe. GEOMETRIC SEQUENCE RECURSIVE DEFINITION Give the f f f r or give a, the a a r where r is called the commo ratio ad ca be positive or egative ad is ofte fractioal. Exercise #5: Geerate the ext three terms of the geometric sequeces give below. (b) f f with (a) a 4 ad r 3 f 9 (c) t t with t 3 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 #6: Cosider a ad a a 3. (a) Geerate the value of a 4. (b) How may times did you eed to multiply by 3 i order to fid a 4. (c) Determie the value of a 0. th (d) Write a formula for the term of a geometric sequece, a, based o the first term, a, r ad. COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #
Name: Date: ARITHMETIC AND GEOMETRIC SEQUENCES COMMON CORE ALGEBRA II HOMEWORK FLUENCY. Geerate the ext three terms of each arithmetic sequece show below. (a) a ad d 4 (b) f f f 8 with 0 (c) a 3, a. I a arithmetic sequece t t 7. If t 5 determie the values of t 4 ad t 0. Show the calculatios that lead to your aswers. 3. If x 4, x5, 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 ad f f 4 the which of the followig represets the value of 40 () 48 (3) 44 () 40 (4) 7 COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON # f? 5. I a arithmetic sequece of umbers a 4 ad a 6 46. Which of the followig is the value of a? () 0 (3) 9 () 46 (4) 06 6. The first term of a arithmetic sequece whose commo differece is 7 ad whose d term is give by a 43 is which of the followig? () 5 (3) 7 () 4 (4) 8
7. Geerate the ext three terms of each geometric sequece defied below. (a) a 8 with r (b) a 3 a ad 6 a (c) f f f ad 5 8. Give that a 5 ad a 5 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 0 9. I a geometric sequece, it is kow that a ad a4 64. The value of a 0 is () 65,536 (3) 5 () 6,44 (4) 4096 APPLICATIONS 0. 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 the this process was allowed to cotiue forever, explai why the perimeter would become ifiitely large. COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #
Name: Date: SUMMATION NOTATION COMMON CORE ALGEBRA II 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 us ed later i the course whe we wat to write formulas used i statistics. ia SUMMATION (SIGMA) NOTATION f i f a f a f a f 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 each time). Exercise #: Evaluate each of the followig sums. (a) 5 i (b) i3 3 k (c) k j j 5 (d) i (e) k (f) ii i k 0 3 i Exercise #: Which of represets the value of () 0 (3) 5 4 i? i () 9 4 (4) 3 4 COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #3
Exercise #3: Cosider the sequece defied recursively by a a a ad a 0 ad a. 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) 9 6 5 00 (b) 6303 5 (c) 5 65 5 5 Exercise #5: Which of the followig represets the sum 3 6 4 48? () 5 3 i (3) i 4 i0 6 i 4 () 3 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. Show your i i i COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #3
Name: Date: SUMMATION NOTATION COMMON CORE ALGEBRA II HOMEWORK FLUENCY. Evaluate each of the followig. Place ay o-iteger aswer i simplest ratioal form. (a) 5 3 4i (b) k i k 0 (c) j 0 j (d) 3 i (e) k i k 0 3 (f) log0 i i (g) 4 (h) 4 i 4 i i i (i) 3 k 0 56 k. Which of the followig is the value of 4k () 53 (3) 37 () 45 (4) 80 4? k 0 3. The sum () 5 8 7 7 i is equal to i4 (3) 3 4 () 3 (4) 7 8 COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #3
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) 4 8 64 8 (b) (c) 4 9 4 44 49 4 9 8 00 5. Which of the followig represets the sum 5 0 8 0? 6 0 (3) j j j () 4j 3 03 (4) 4 j () j j3 j0 6. A sequece is defied recursively by the formula b 4b b with b ad b 3. What is the value 5 of bi? Show the work that leads to your aswer. i3 REASONING 6. A curious patter occurs whe we look at the behavior of the sum k (a) Fid the value of this sum for a variety of values of below: k 4: k : k 4 k. k 3 k 5: k 3: k 5 k (b) What types of umbers are you summig? What types of umbers are the sums? 96. (c) Fid the value of such that k k COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #3
Name: Date: ARITHMETIC SERIES COMMON CORE ALGEBRA II 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: 3 S i a i I truth, you have already worked extesively with series i previous lessos almost aytime you evaluated a summatio problem. Exercise #: Give the arithmetic sequece defied by a ad a a 5, the which of the followig is the value of S 5 5 a? i () 3 (3) 5 () 40 (4) 7 i 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 #: Cosider the arithmetic sequece defied by a 3 ad a a. 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 7 935 7 (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, its last term, a, ad the umber of terms,. COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #4
SUM OF AN ARITHMETIC SERIES Give a arithmetic series with terms, a, a,..., a, the its sum is give by: S a a Exercise #3: Which of the followig is the sum of the first 00 atural umbers? Show the process that leads to your choice. () 5,000 (3) 0,000 () 5,00 (4) 5,050 Exercise #4: Fid the sum of each arithmetic series described or show below. (a) The sum of the sixtee terms give by: 0 6 46 50. (b) The first term is 8, the commo differece, d, is 6 ad there are 0 terms (c) The last term is a 9 ad the commo differece, d, is 3. (d) The sum 5 8 77. Exercise #5: Kirk has set up a college savigs accout for his so, Maxwell. If Kirk deposits $00 per moth i a accout, icreasig the amout he deposits by $0 per moth each moth, the how much will be i the accout after 0 years? COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #4
Name: Date: ARITHMETIC SERIES COMMON CORE ALGEBRA II HOMEWORK FLUENCY. Which of the followig represets the sum of 3087 94 if the arithmetic series has 4 terms? (),358 (3) 679 () 658 (4),76. The sum of the first 50 atural umbers is (),75 (3),50 (),875 (4) 950 3. If the first ad last terms of a arithmetic series are 5 ad 7, respectively, ad the series has a sum 9, the the umber of terms i the series is () 8 (3) 4 () (4) 4. Fid the sum of each arithmetic series described or show below. (a) The sum of the first 00 eve, atural umbers. (b) The sum of multiples of five from 0 to 75, iclusive. (c) A series whose first two terms are ad 8, respectively, ad whose last term is 4. (d) A series of 0 terms whose last term is equal to 97 ad whose commo differece is five. COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #4
5. For a arithmetic series that sums to,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 4 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 icreasig his deposit by $5 per moth each moth after. If he saves this way for the ext 0 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 6 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 0 secods? Show the work that leads to your aswer. 9. A large gradfather clock strikes its bell oce at :00, twice at :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. COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #4
Name: Date: GEOMETRIC SERIES COMMON CORE ALGEBRA II 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 #: Give a geometric series defied by the recursive formula a 3 ad a a, which of the followig is the value of S 5 5 a? i () 06 (3) 93 () 75 (4) 35 i The sum of a fiite umber of geometric sequece terms is less obvious tha that for a arithmetic series, but ca be foud oetheless. The ext exercise derives the formula for fidig this sum. Exercise #: Recall that for a geometric sequece, the th term is give by form of a geometric series is give below. S a arar ar ar a a r. Thus, the geeral (a) Write a expressio below for the product of r ad S. (b) Fid, i simplest form, the value of S r S i terms of a, r, ad. rs S rs (c) Write both sides of the equatio i (b) i their factored form. (d) From the equatio i part (c), fid a formula for S i terms of a, r, ad. Exercise #3: 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? () 3,756 (3) 4,560 () 8,765 (4) 65,535 COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #5
SUM OF A FINITE GEOMETRIC SERIES For a geometric series defied by its first term, a, ad its commo ratio, r, the sum of terms is give by: a r a ar S or S r r Exercise #4: Fid the value of the geometric series show below. Show the calculatios that lead to your fial aswer. 6 4 768 Exercise #5: 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 0 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 0 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 th year. (c) Based o (b), write a geometric sum represetig the amout of moey i Maria s accout after 0 years. (d) Evaluate the sum i (c) usig the formula above. Exercise #6: A perso places pey i a piggy bak o the first day of the moth, peies o the secod day, 4 peies o the third, ad so o. Will this perso be a millioaire at the ed of a 3 day moth? Show the calculatios that lead to your aswer. COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #5
Name: Date: GEOMETRIC SERIES COMMON CORE ALGEBRA II HOMEWORK FLUENCY. Fid the sums of geometric series with the followig properties: (a) a 6, r 3 ad 8 (b) a 0, r, ad 6 (c) a 5, r, ad 0. If the geometric series () 48 7 () 74 3 8 54 36 has seve terms i its sum the the value of the sum is 7 (3) 370 9 (4) 84 54 3. A geometric series has a first term of 3 ad a fial term of this series is () 9.75 (3).5 () 6.5 (4).5 ad a commo ratio of 4. The value of 4. Which of the followig represets the value of series has i it. () 9,7 (3),34 8 i0 i 3 56? Thik carefully about how may terms this (),60 (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 () 8 (3) 6 () 5 (4) 4 COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #5
6. Fid the sum of the geometric series show below. Show the work that leads to your aswer. 7 9 3 79 APPLICATIONS 7. I the picture show at the right, the outer most square has a area of 6 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 $000 is placed the accout o Jauary 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 $000 is deposited at the begiig of the 9 th year? Show the sum that leads to your aswer as well as relevat calculatios. 9. A ball is dropped from 6 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 4 ball has traveled whe it strikes the groud for the 0 th time? Write out the first five terms of this sum to help visualize. COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #5
Name: Date: MORTGAGE PAYMENTS COMMON CORE ALGEBRA II Mortgages, ot just o houses, are large amouts of moey borrowed from a bak o which iterest is calculated (added o) o a regular (typically mothly) basis. Regular paymets are also 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 ultimately ivolves geometric series. First, some basics. Exercise #: Let's say a perso takes out a mortgage for $00,000 ad wats to make paymets of $600 each moth of pay it off. The bak is goig to charge this perso 4% omial yearly iterest, applied mothly. (a) What is the amout owed at the ed of the first moth? Show the calculatios that lead to your aswer. (b) How much of the first moth's paymet wet to payig off the pricipal? How much of it wet to payig iterest o the loa? Show your calculatios. Amout Towards Pricipal Amout Towards Iterest (c) Determie the amout owed at the ed of the secod moth. Agai, show the calculatios that lead to your aswer. (d) The amout owed at the ed of a moth actually forms a sequece that ca be defied recursively. If a 00, 000, the defie a recursive rule that gives this sequece. Exercise #: If a perso took out a $50,000 mortgage at 5% yearly iterest, why would it be uwise to have mothly paymets of $500? COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #6
Now, eve if we ca defie the sequece recursively, as i Exercise #(d), it would be ice to have a formula that would calculate what we owed after a certai umber of moths explicitly. To do this, we must see a tricky, exteded patter. Exercise #3: Let's go back to our example of the $00,000 mortgage at 4% yearly iterest. Remember, we are payig off this mortgage with $,600 mothly paymets (much of which are iitially goig to iterest). Let's see if we ca determie how much we still owe after -paymets. To make our work easier to follow (ad more.04 geeral), we will let r.003, P 00,000, ad m, 600 to stad for mothly rate (i decimal form, our pricipal, ad our mothly paymet). (a) Explai what the followig calculatio represets. P r m (b) What does the followig calculatio represet? Write it i expaded form, but leave the biomial r. P r m r m (c) Based o (a) ad (b), cotiue this lie of thikig to write expressios for the amout owed at the ed of 3 moths ad 4 moths. (d) Based o (c), write a equatio for how much would be owed after moths. Exercise #4: Now let's see the geometric series. I your aswer to (d), you should have the followig expressio: 3 m r m r m r m r m r m 3 Give that this is equivalet to: m m r m r m r m r, fid the sum of the geometric series iside of the paretheses. Thik carefully about the umber of terms i this expressio. Exercise #5: Fid the amout owed o this loa after 0 years or 0 paymets (moths). COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #6
Name: Date: MORTGAGE PAYMENTS COMMON CORE ALGEBRA II HOMEWORK APPLICATIONS. Cosider the mortgage loa of $50,000 at a omial 6% yearly iterest applied mothly at a rate of 0.5% per moth. Mothly paymets of $,000 are beig made o this loa. (a) Determie how much is owed o this loa at the ed of the first, secod, third moth ad fourth moths. Show the work that leads to your aswers. Evaluate all expressios. Oe Moth: Two Moths: Three Moths: Four Moths: (b) The amouts that are owed at the ed of each moth form a sequece that ca be defied recursively. Give that a $50, 000 represets the first amout owed, give a recursive rule based o what you did i (a) that shows how each successive amout owed depeds o the previous oe. (c) Usig a geometric series approach (i.e. the formula we developed i Exercises #3 ad #4), determie how much is still owed after 5 years of paymets. Show your work. (d) Will this loa be paid off after 0 years? What about 30? Provide evidece to support both aswers. COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #6
Whe a loa officer speaks to people about a loa for a certai pricipal, P, at a certai mothly rate, r, they always have to balace two quatities, the mothly paymet, m, with the umber of paymets,, it takes to pay off the loa. These two vary iversely. All of these quatities ca be related by the formula: P r m r This formula is derived by takig the formula we arrived at i Exercises #3 ad #4 ad settig what we owe equal to zero.. Calculate the mothly paymet eeded to pay off a $00,000 loa at 4% yearly iterest over a 0 year period. Recall that r is the mothly rate. Show your work ad carefully evaluate the above formula for m. 3. Do the same calculatio as i the previous exercise but ow make the pay off period 30 years istead of 0. How much less is your mothly paymet? It is of iterest to also be able to calculate the umber of paymets (ad hece the pay off period) if you have a mothly paymet i mid. But this is much more difficult give that you must solve for. 4. Give the formula above: (a) Show that P r log m log r (b) Usig (a), determie the umber of moths it would take to pay off a $50,000 loa at a mothly 0.5% rate with $,000 paymets. COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES LESSON #6