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Lesson 12.1: Intro to Sequence and RecursionLearning Goals:What is a sequence? What notation do we use?What is a recursive sequence and how can we find the nthterm of it?Sequence: a list of numbers or objects in a specific orderExample 1: 3, 7, 11, 15, 19, 23, …Example 2: 0, 2, 0, 2, 0, 2, …Example 3: Terms: each member of a sequence3, 7, 11, 15, 19, 23, …0, 2, 0, 2, 0, 2, …1st term =31st term =01st term =14th term =154th term =24th term =7-63546990SequencesA sequence is a function whose domain is a set of positive integers. If a domain is not specified, it is understood that the domain starts with 1. The values in the range are called the terms of the sequence.A finite sequence has a limited number of terms. n=1 2 3 4An infinite sequence continues without stopping.an=2 4 6 8Finite sequence: 2, 4, 6, 8Infinite sequence: 2, 4, 6, 8, …A sequence can be specified by an equation, or rule. For example, both sequences above can be described by the rule an=2n or fn=2na3=23=6 or f4=24=8 020000SequencesA sequence is a function whose domain is a set of positive integers. If a domain is not specified, it is understood that the domain starts with 1. The values in the range are called the terms of the sequence.A finite sequence has a limited number of terms. n=1 2 3 4An infinite sequence continues without stopping.an=2 4 6 8Finite sequence: 2, 4, 6, 8Infinite sequence: 2, 4, 6, 8, …A sequence can be specified by an equation, or rule. For example, both sequences above can be described by the rule an=2n or fn=2na3=23=6 or f4=24=8 Example 1: Write the first six terms of an=2n+1 the formula is the rule for the sequence!n=1 n=2 n=3 n=4 n=5 n=621+1 22+1 23+1 24+1 25+1 26+1a1=4 a2=8 a3=16 a4=32 a5=64 a6=128{4, 8, 16, 32, 63, 128} Example 2: Write the first 4 terms of the sequence: an=1+-1nnn=1n=2n=3n=41+-111 1+-1221+-1331+-144a1=0 a2=1a3=0a4=120, 1, 0, 12 Find the indicated term of the sequence:3. an=-1n3n-24. fn=4n2n2-3a25=-1253(25)-2f11=4(11)2(11)2-3a25=(-1)(75-2)f11=442121-3a25=-73f11=44239Writing the Rule for SequencesDescribe the pattern, write the next term, and write a rule for the nth term of the sequence {2, 4, 6, 8, …} {2, 3, 4, 5,…}n=1 2 3 4 n=1 2 3 4 ×2↓ ×2↓ ×2↓ ×2↓ +1↓ +1↓ +1↓ +1↓ an=2 4 6 8 an=2 3 4 5 an=2n or fn=2n an=n+1 or fn=n+1 5. Describe the pattern, write the next term, and write a rule for the nth term of the sequence.a) 1, 4, 9, 16, …b) 0, 7, 26, 63, …n=1 2 3 4 n=1 2 3 4 ^2↓ ^2↓ ^2↓ ^2↓ ↓ ↓ ↓ ↓ an=1 4 9 16 an=0 7 26 63 an=n2 or fn=n2 an=n3-1or fn=n3-1 c) 13, 1 9, 127, 181, …n=1 2 3 4 an=13n or fn=13n 6. The diagrams below represent the first three terms of a sequence. Assuming the pattern continues, which formula determines an, the number of shaded squares in the nth term?(1) an=4n+12 (2) an=4n+8 (3) an=4n+4 (4) an=4n+2Vocabulary:Explicit Rule: a formula that defines a sequence2, 4, 6, 8, … (sequence)an=2n or fn=2n (explicit rule)Recursive Rule: a formula that requires the computation of the previous term in order to find the next term of a sequence. (1) starting term (2) rule that uses previous term445770044577000The example below is a recursive sequence because you need the first term to find the second term.a1=-4an=an-1+5a1=-4 a2=a2-1+5=a1+5=-4+5=1 a3=a3-1+5=a2+5=1+5=6 a4=a4-1+5=a3+5=6+5=11 Finding Terms of Recursive Sequences4086860316230007. Write the first four terms of the recursive sequence given below:a1=2 an=an-1-3a1=2 a2=a2-1-3=a1-3=2-3=-1 a3=a3-1-3=a2-3=-1-3=-4 a4=a4-1-3=a3-3=-4-3=-7 3752850459740008. Write the first five terms of the recursive sequence given below:f1=4 fn=2f(n-1)f1=4 f2=2f2-1=2f1=24=8 f3=2f3-1=2f2=28=16 f4=2f4-1=2f3=216=32 f5=2f5-1=2f4=232=64 9. For the recursively defined sequence tn=tn-12+2 and t1=2 the value of t4 is (1) 18 (2) 38 (3) 456 (4) 1446t1=2 t2=t2-12+2=t12+2=22+2=4+2=6 t3=t3-12+2=t22+2=62+2=36+2=38 t4=t4-12+2=t32+2=382+2=1444+2=1446 Homework 12.1: Intro to Sequence and Recursion1. Given the sequence 7, 11, 15, 19, …, which of the following represents a formula that will generate it?(1) an=4n+7 (2) an=3n+4 (3) an=3n+7 (4) an=4n+32. Determine and label the first four terms of each of the following sequences.(a) an=n2-5(b) tn=23n(c) tn=1n+13. Which of the following formulas would represent the sequence 10, 20, 40, 80, 160, …(1) an=10n (2) an=10(2)n (3) an=5(2)n (4) an=2n+104. Write an algebraic expression that represents anfor each sequence.(a) 3, 9, 27, 81, …(b) 12, 13, 14, 15, …(c) 1, 2, 3, 2, …5. A recursive sequence is defined by an+1=2an-an-1 with a1=0 and a2=1. Which of the following represents the value of a5? (1) 8 (2) -7 (3) 3 (4) 46. Sequences below are defined recursively. Determine and label the next three terms of the sequence.(a) a1=5, an=an-1+1(b) a1=-2, an=-2an-1(c) bn=bn-1+2n with b1=57. Sean has started an exercise program. The first day he worked out for 30 minutes. Each day for the next six days, he increased his time by 5 minutes.(a) Write a sequence for the number of minutes that Sean worked out for each of the seven days.(b) Write a recursive definition for this sequence.8. A pattern of blocks is shown below. If the pattern of blocks continues, which formula(s) could be used to determine the number of blocks in the nth term?IIIIIIan=n+4a1=2an=an-1+4an=4n-2(1) I and II (2) I and III (3) II and III (4) III, only12.2: Arithmetic and Geometric SequencesLearning Goals:What is an arithmetic sequence? What is the common difference of an arithmetic sequence?How can we find the nth term of an arithmetic sequence?What is a geometric sequence? What is the common ratio of a geometric sequence?How can we find the nth term of a geometric sequence?Arithmetic Sequence: a sequence where you are adding a fixed amount from one term to the next1, 3, 5, 7, 9, … adding 2 4, 2, 0, -2, -4 subtracting 2Common Difference: (d=a2-a1) the fixed amount added to get to the next term. 1, 3, 5, 7, 9, … d= 2 4, 2, 0, -2, -4 d=- 2d=3-1=2 or d=9-7=2 d=2-4=-2 or d=-4--2=-2Identify Arithmetic Sequences:Example 1: Tell whether the sequence -5, -3, -1, 1, 3, … is arithmetic.48006008128000Find the differences of consecutive terms. a2-a1=-3--5=2 a3-a2= -1--3=2a4-a3=1--1=2 a5-a4= 3-1=2Arithmetic because you add 2 to get the next term! Arithmetic sequences can be represented by a linear scatter plot.D=-5, -3, -1, 1, 3, … R=2, 4, 6, 8, …0180975RULE FOR AN ARITHMETIC SEQUENCEThe nth term of an arithmetic sequence with the first term a1 and common difference d is given by: an=a1+n-1d On Reference Sheet!00RULE FOR AN ARITHMETIC SEQUENCEThe nth term of an arithmetic sequence with the first term a1 and common difference d is given by: an=a1+n-1d On Reference Sheet!Example 2: Write a rule for the nth term of the sequence. Then find a19a. 2, 9, 16, 23, …a1=2 Common Difference: d=a2-a1=9-2=7 or a3-a2=16-9=7an=a1+n-1d an=7n-5 an=2+n-17 a19=719-5 an=2+7n-7 a19=128 an=7n-5 b. 57, 45, 33, 21, …a1=57 Common Difference: d=a2-a1=45-57=-12 or a3-a2=33-45=-12an=a1+n-1d an=-12n+69 an=57+n-1(-12) a19=-1219+69 an=57-12n+12 a19=-159 an=-12n+69 Example 3: One term of an arithmetic sequence is a11=41. The common difference is d=5. Write a rule for the nth term.a11=41 so n=11 d=5 a1=? an=a1+n-1d an=a1+n-1d a11=a1+11-15 an=-9+n-15 41=a1+105 oran=-9+5n-5 41=a1+50 oran=5n-14a1=-9 Geometric Sequence: a sequence where you are multiplying by a fixed amount from one term to the next2, 6, 18, 54, … multiplying by 3 18, 6, 2… multiplying by 13Common Ratio: r=a2a1the fixed amount multiplied to get to the next term2, 6, 18, 54, … r= 3 18, 6, 2… r= 13r=62=3 or 5418=3 r=618=13 or 26=1344196004381500Geometric because you multiply to get the next term! Geometric sequences can be represented by an exponential scatter plot.D=1, 2, 3, 4, … R=1, 2, 4, 8, …Identify Geometric Sequences:Example 1: Tell whether the sequence 1, -4, 16, -64, 256, … is geometric.To decide whether a sequence is geometric, find the ratios of consecutive terms.a2a1=-41=-4 a3a2=16-4=-4 a4a3= -6416=-4 a5a4= 256-64=-4Geometric because you multiply by -4 to get to the next term.304165180975RULE FOR A GEOMETRIC SEQUENCEThe nth term of a geometric sequence with the first term a1 and common ratio r is given by: an=a1 rn-1 On Reference Sheet!020000RULE FOR A GEOMETRIC SEQUENCEThe nth term of a geometric sequence with the first term a1 and common ratio r is given by: an=a1 rn-1 On Reference Sheet!Example2: Write a rule for the nth term of the sequence 972, -324, 108, -36, … Then find a10.Geometric because you multiply by -13. r=-324972=-36108=-13an=a1 rn-1 an=972 -13n-1 a10=972 -1310-1=972-139=-481 Example 3*: One term of a geometric sequence is a3=-18. The common ratio is r=3. Write a rule for the nth term.an=a1 rn-1 an=a1 rn-1 a3=a1 33-1 an=(-2) (3)n-1 -18=a1 32 -18=9a1 a1=-2 Homework 12.2: Arithmetic and Geometric Sequences1. Does the following pair of formulas represent the same sequence? Justify your reasoning.an+1=23an, a0=-1, and bn=-23n, for n≥02. A sequence has the following terms: a1=4, a2=10, a3=25, a4=62.5. Which formula represents the nth term in the sequence?(1) an=4+2.5n (2) an=4+2.5(n-1) (3) an=4(2.5)n (4) an=4(2.5)n-13. Which arithmetic sequence has a common difference of 4?(1) {0, 4n, 8n, 12n, …} (2) {n, 4n, 16n, 64n, …} (3) {n+1, n+5, n+9, n+13, …} (4) {n+4, n+16, n+64, n+256, ….}4. An arithmetic sequence has a first term of 10 and a sixth term of 40. What is the 20th term of this sequence?(1) 105 (2) 110 (3) 124 (4) 1305. What is the fourth term of the sequence defined by a1=3xy5 and an=2xyan-1?(1) 12x3y3 (2) 24x2y4 (3) 24x4y2 (4) 48x5y6. What is the common ratio of the sequence 164a5b3,- 332a3b4, 916ab5,…?(1) -3b2a2 (2) -6ba2 (3) -3a2b (4) -6a2b7. What is the common ratio of the geometric sequence whose first term is 27 and the fourth term is 64?(1) 34 (2) 6481 (3) 43 (4) 373Lesson 12.3: Recursive Sequences vs. Explicit FormLearning Goal: How do we write a recursive sequence in explicit form and vice versa?Write the first four terms of the recursive sequence given below:a1=-1 an=2an-1+na2=2a2-1+2=2a1+2=2-1+2=-2+2=0 a3=2a3-1+3=2a2+3=2(0)+3=0+3=3 a4=2a4-1+4=2a3+4=23+4=6+4=10 44767502667000Notation: Recursive forms work with the term(s) immediately in front of the term being examined. The table shows that there are many options as to how this relationship may be expressed in notations.A recursive formula is written with two parts: a statement of the first term along with a statement of the formula relating successive terms.481965087058500The statements below are all naming the same sequence:a1=1; an=an-1+4a1=1; an-1=an+4f1=1; fn=fn-1+4f1=1; fn+1=fn+4*For multiple choice questions, you need to know that all of these mean the same thing! Rewrite the given sequence 3 other ways, using the notation in the above example.a1=-6a1=2an+1=an+5an=3an-1f1=-6 f1=2fn+1=fn+5 fn+1=3f(n)f1=-6 f1=2fn=fn-1+5 fn=3f(n-1)a1=-6 a1=2an=an-1+5 an+1=3anArithmetic! Adding 5Geometric! Multiply by 3Converting from Recursive to ExplicitUse the following explicit formulas with a1 as the first term (Use for n≥1)ARITHMETIC SEQUENCE:an=a1+n-1dGEOMETRIC SEQUENCE:an=a1rn-1Example 1: Convert each of the following recursive formulas to explicit formulas. Identify each sequence as arithmetic, geometric, or neither.a. an=4an-1 with a1=12Geometric because you multiply by 4 to get the next term!a1=12 & r=4an=a1rn-1=12(4)n-1 b. an=4.2+an-1 with a1=12Arithmetic because you add 4.2 to get the next term!a1=12 & d=4.2an=a1+n-1d=12+(n-1)(4.2) c. an+1=5 an with a1=2Geometric because you multiply by 5 to get the next term!a1=2 & r=5an=a1rn-1=2(5)n-1 d. an+1=5+an with a1=2Arithmetic because you add 5 to get the next term!a1=2 & d=5an=a1+n-1d=2+(n-1)(5) Convert from Explicit to RecursiveUse the following recursive formulas (MEMORIZE)ARITHMETIC SEQUENCE:Subscript notation:a1=x and an=an-1+dFunction notation:f1=x and fn=fn-1dGEOMETRIC SEQUENCE:Subscript notation:a1=x and an=an-1rFunction notation:f1=x and fn=r?f(n-1)Example2: Write each sequence in recursive form.a. an=153n for n≥0 so use n=0 for the 1st term a0a0=1530=15 a1=1531=35 a2=1532=95 Geometric because you multiply by 3 to get the next term!a0=15 & r=3an=3an-1 b. an=16-2n for n≥1 so use n=1 for the 1st term a1a1=16-21=14 a2=16-22=12 a3=16-23=10 Arithmetic because you add -2 to get the next term!a1=14 & d=-2an=an-1-2 c*. an=1612n for n≥1 so use n=1 for the 1st term a1a1=16121=8 a2=16122=4 a3=16123=2 Geometric because you multiply by 12 to get the next term! a1=8 & r=12an=12an-1 d*. an=71-67n for n≥0 so use n=0 for the 1st term a0a0=71-670=71=4977 a1=71-671=4917 a2=71-672=4857 Arithmetic because you add -67 to get the next term!a0=71 & d=-67an=an-1-67 Example 3: At her job, Pat earns $25,000 the first year and receives a raise of $1,000 each year. The explicit formula for the nth term of this sequence is an=25,000+n-11000. Write a rule to represent the equivalent recursive formula of this sequence.a1=25000an=an-1+1000Example 4: The initial value of a specialty toy is $3,000, and its value each of the following years is 20% less. Write an explicit formula and a recursive formula to model this sequence. 20% less means they have 80% lefta1=3000r=0.80 explicit: an=3000(0.80)n-1Recursive: a1=3000an=0.80(an-1)Example 5: The last row of an auditorium has 2500 seats, and each row before it has 80 fewer seats than the row behind it. Write an explicit formula and a recursive formula to model the sequence.explicit: an=2500+(n-1)(-80)Recursive: a1=2500an=an-1-80Homework 12.3: Recursive Sequences vs. Explicit Form1. Write each recursive sequence in explicit form. Identify each sequence as arithmetic, geometric, or neither.a. a1=3, an+1=an+5b. a1=2, an=an-122. Which of the following formulas is equivalent to the geometric sequence modeled by the equation an=52n-1?(1) a1=5, an+1=2an(3) a1=10, an+1=2an(2) a1=2, an+1=5an(4) a1=5, an+1=an23. If a geometric sequence has a1=256 and a8=512, find the exact value of the common ratio r.4. Given the sequence 7, 11, 15, 19, … write an explicit formula to represent this sequence.5. Which recursively defined function has a first term equal to 10 and a common difference of 4?(1) f1=10, fx=fx-1+4(2) f1=4, fx=fx-1+10(3) f1=10, fx=4f(x-1)(4) f1=4, fx=10f(x-1)6. The recursive formula, a2=18.5, an=an-1+1.5 models the number of people, in millions, that own a smartphone in the U.S. n years after 2014.a. Write an equivalent explicit formula for the situation.b. Find the value of a5 using both the recursive formula and the explicit formula. Explain what a5 means in the context of the problem.7. In an arithmetic sequence of numbers a1=-4 and a6=46. Find the value of a12.8. Given that a1=5 and a2=15 are the first two terms of a geometric sequence, determine the values of a3 and a10. Show the calculations that lead to your answers.9. The formula below can be used to model which scenario?a1=2000an+1=an+50The last row of a concert hall has 2000 seats and each row before has 50 less seats than the one after it.The first row of a concert hall has 2000 seats and each row after has 50 more seats than the previous row.A bank account starts with a deposit of $2000 and it grows each year by 50%.A used car is worth $2000 to start and depreciates by 50% each subsequent year.Lesson 12.4: Summation NotationLearning Goals:What is summation notation? How can we evaluate the sum of a finite series given in summation notation?How can we write a series in summation notation?Summation NotationThe summation of a specific number of terms of a sequence (a series) can also be represented in a compact form, called summation notation, or sigma notation.The Greek capital letter sigma, , is used to indicate a sum.Given a sequence a1, a2, a3, a4, … we can write the sum of the first n terms of the sequence using the expression:k=1nak=a1+a2+a3+…+anIt is read, “The sum of ak from k=1 to k=n.Example: n=152n=21+22+23+24+25=2+4+6+8+10=30To write the terms of the series, replace n by the consecutive integers from 1 to 5, as shown above.2nd→ stat →math →sum(5) & 2nd →stat →ops →seq 2x, x, 1, 5Example 1: Write the sum without using summation notation. You do not need to find the sum.k=05k=0+1+2+3+4+5 j=56j2=52+62+72i=241i=12+13+14i is not an imaginary unit in this example!j=16jxj-1=1x1-1+2x2-1+3x3-1+4x4-1+5x5-1+6x6-1=1+2x+3x2+4x3+5x4+6x5Example 2: Write the sum without using summation notation, and find the sum.m=042m+xm=20+x0+21+x1+22+x2+23+x3+24+x4= 0+1 + 2+x + 4+x2 + 6+x3 + 8+x4=21+x+x2+x3+x4j=03-4j-2=-40-2+-41-2+-42-2+-43-2=-4-2+-4-1+-40+-41=1-42+1(-4)+1-4=116+1-4-3=-5116p=13p?3p=102 math→0:summationWhat if… 3p=13p?3p=3102=306 just multiply the answer by 3 k=14logkk+1=-.6987What if… 3+k=14logkk+1=3-.6987=-2.3010 just add 3 to the final sumExample 3: Write each sum using summation notation.Strategies:See if sequence is arithmetic or geometric→if it is you can find the explicit formula for the sequenceIf the sequence is not arithmetic or geometric, try to look for other patternsLook to see if numbers are “famous” numbers like perfect squares or cubesk=1nak2+4+8+16+…+512a1=2 & r=2an=a1rn-1 512=22n-1 256=2n-1 get like “bases”k=192(2)k-128=2n-1 8=n-1 n=9 Example 4: What is the value of 3n=26n2?(1) 10 (2) 13 (3) 30 (4) 60Homework 12.4: Summation Notation1. Write the sum without using summation notation, and find the sum.a. j=031052j+1b. k=05(-1)kxkc. m=141632m2. Which of the following represents the sum of a geometric sequence?(1) j=152j+2j (2) k=2623k-1 (3) p=36(3p+1) (4) m=262m23. Write each sum using summation notation.a. 2+4+6+8+…+100b. 12+22+32+42+…+100002c. 11?2+12?3+13?4+…+149?50Lesson 12.5: Geometric SeriesLearning Goal: What is a geometric series and how do we evaluate it?Do Now: Answer the following questions in order to prepare for today’s lesson.7524758255Geometric Sequence: an=a1?rn-1020000Geometric Sequence: an=a1?rn-11. Given the geometric sequence: -375, -75, -15, -3, …a. Write the formula for the nth term of the geometric sequence.r=-75-375=15 an=-37515n-1b. Use the formula to find the 7th term of the sequence. n=7a7=-375157-1=-.024=-3125 c. Find the sum of the first 7 terms of this sequence. -375, -75, -15, -3, -35, -325, -3125 sum =-468.744OR Sn=a1-a1rn1-r=-375-(-375)1571-15=-468.744 GEOMETRIC SERIES (on Reference Sheet)To find the sum of a certain number of terms of a geometric sequence:Sn=a1-a1rn1-r Where Sn= the sum of n terms, a1= the first term, and r= the common ratio1. Find the sum of the first 8 terms of the sequence: -5, 15, -45, 135, …Why is this a geometric sequence/series? Because you are multiplying by -3Sn=a1-a1rn1-r S8=-5--5-381-(-3)=8200 2. Given the geometric series: -6+9-272+… find the sum of the first 11 terms.How do you find the common ratio? -6, 9, -272, …r=9-6=-1.5Sn=a1-a1rn1-r=-6-(-6)-1.5111-(-1.5)=-107517512 3. Find the sum of the following summation notations by using the geometric series formula.k=1104(3)k-1 an=a1rn-1 form so it is geometric!Sn=a1-a1rn1-r=4-4(3)101-3=118096 GEOMETRIC SERIES (need to find number of terms)If you need to find the number of terms in a geometric series, use the geometric sequence formula to find "n".GEOMETRIC SEQUENCE:an=a1rn-1GEOMETRIC SERIESSn=a1-a1rn1-r4. Calculate the sum of the geometric series: 4+12+…+972.an=a1rn-1 Sn=a1-a1rn1-r972=4(3)n-1 Sn=4-4(3)61-3243=3n-1 Sn=145635=3n-1 5=n-1 n=6 5. Calculate the sum of the geometric series: 1-12+14-18+…-1512an=a1rn-1 Sn=a1-a1rn1-r-1512=1-12n-1 Sn=1-1(-12)101-(-12)-1512=-12n-1 Sn=341512-129=-12n-1 9=n-1 n=10 6. Calculate the sum of the geometric series: 1+3+3+33+…+243S10=1-1-12101--12=3415127*. The sum of a geometric series with four terms is 60, and the common ratio is r=12. Find the first term.Sn=a1-a1rn1-r=a1(a-rn)1-r Sn=60, n=4, r=1260=a11-1241-1230=a1(1-116)a1=32Homework 12.5: Geometric Series1. Find the sum of the first 8 terms of the sequence: -12, -48, -192, -768, …2. Find the sum i=1126(-2)i-1 by using the geometric series formula.3. Calculate the sum of the geometric series: 0.8+0.64+0.512+…+0.327684. A geometric sequence has all its terms positive. The first term is 810 and the fifth term is 10. Find the common ratio.5. The sum of a geomtric series with seven terms is -10,922, and the common ratio is r=4, Find the first term.6. The sixth term of an arithmetic sequence is 24. The common difference is 8. Calculate the first term of the sequence.Lesson 12.6: Geometric Series Word ProblemsLearning Goal: How do we use the geometric series formula when working with word problems?Do Now: Answer the following question in order to prepare for today’s lesson.1. What are the formulas for exponential growth and decay? What do the variables represent in each formula?Growth: A=P1+rt A=final amount, P=initial amount, r=rate, t=timeDecay: A=P1-rt A=final amount, P=initial amount, r=rate, t=time2. A computer valued at $6500 depreciates at the rate of 14.3% per year.a. Write a function that models the value of the computer after n years.Decay: A=P1-rt Geometric!A=6500(1-.143)n an=a1rn-1A=6500(.857)n b. Find the value of the computer after three years. n=3A=6500(.857)3=$4091.25 3. A geometric sequence has a first term of 36 and a common ration of 13.a. Write a geometric series formula, Sn, to represent this sum over n terms.Sn=a1-a1rn1-r=36-3613n1-13 b. Use this formula to find the sum of the first 15 terms.S15=36-3613151-13=53.99999624 GEOMETRIC SEQUENCES WITH PERCENTAGES-95250506730GEOMETRIC SEQUENCE FORMULATo find any term of a geometric sequence:an=a1rn-1Where an= the nth term, a1= the first term, and r= the common ratio020000GEOMETRIC SEQUENCE FORMULATo find any term of a geometric sequence:an=a1rn-1Where an= the nth term, a1= the first term, and r= the common ratioIf a question refers to a percent, this means you are dealing with a geometric sequence.When given a percent, the common ratio is the percent remaining of the previous term.Growth: A=P1+rt & Decay: A=P1-rt Example: Identify the common ratio in each situation.a. A certain water filtration system can remove 70% of the contaminants each time a sample of water is passed through it. Decay!1-r=1-.70=.30=rb. John’s salary earns an increase of 4% each successive year. Growth!1+r=1+.04=1.04=rc. A basketball is dropped vertically. The height of each subsequent bounce is 90% of the previous bounce. Decay! Decreases by 10%1-r=1-.10=.90=r1. A fan is running at 10 revolutions per second. After it is turned off, its speed decreased at a rate of 75% per second. Find an explicit formula for the sequence that represents the number of revolutions after n seconds. Geometrican=a1rn-1 a1=10 r=1-.75=.25an=10.25n-1 2. Suppose you drop a tennis ball from a height of 15 feet. After the ball hits the floor, it rebounds to 85% of its previous height. Write an explicit formula for the sequence. Geometric Decrease by 15%an=a1rn-1 a1=15 r=1-.15=.85an=15.85n-1 GOMETRIC SERIES WORD PROBLEMSGEOMETRIC SERIESSn=a1-a1rn1-rSn= the sum of n terms, a1= the first term, and r= the common ratio3. George has taken a job with a starting salary of $50,000 and receives an annual raise of 2%. Write a geometric series formula, sn, for George’s total earnings over n years.Sn=a1-a1rn1-ra1=50000 r=1+.02=1.02 n=?Sn=50000-50000(1.02)n1-1.02 Use this formula to find George’s total earnings for his first 12 years of working, to the nearest cent.Sn=50000-500001.02121-1.02=$670604.49 4. The first swing of a pendulum travels 40 centimeters. Each subsequent swing travels 95% as far as the previous swing. Write a geometric series formula, sn, for the pendulum’s total distance over n swings. It decreases by 5%Sn=a1-a1rn1-r a1=40 r=1-.05=.95 n=?Sn=40-40(.95)n1-.95 Use this formula to find the pendulum’s total distance after the 30th swing, to the nearest tenth of a centimeter.Sn=40-40(.95)301-.95=628.3 PUTTING IT ALL TOGETHER5. A car with an original price of $30,000 depreciates by 30% each year.a. Write an explicit formula for the price of the car after n years. Decreases by 30%an=a1rn-1=30000(.70)n-1 b. Write a recursive formula, the price of the car after n years.a1=30000 an=0.7an-1 c. Write a geometric series formula, sn, for the car’s total price over n years.Sn=30000-30000(.70)n1-.70 6. Kristin wants to increase her running endurance. According to experts, a gradual mileage increase of 10% per week can reduce the risk of injury. If Kristin runs 8 miles in week one, which expression can help her find the total number of miles she will have run over the course of her 6-week training program?(1) n=168(1.10)n-1 (2) n=168(1.10)n (3) 8-8(1.10)60.90 (4) 8-8(0.10)n1.101+.10=1.10Homework 12.6: Geometric Series Word Problems1. Alexa earns $33,000 in her first year of teaching and earns a 4% increase in each successive year. Write a geometric series formula, Sn, for Alexa’s total earnings over n years. Use this formula to find Alexa’s total earnings for her first 15 years of teaching, to the nearest cent.2. The tuition fees for the first three years of high school are given in the table below. These tuition fees form a geometric sequence.(a) Find the common ratio, r, for this sequence.(b) If fees continue to rise at the same rate, calculate (to the nearest dollar) the total cost of tuition fees for the first six years of high school.3. A basketball is dropped vertically. It reaches a height of 2 cm on the first bounce. The height of each subsequent bounce is 90% of the previous bounce. (a) What height does it reach on the 8th bounce?(b) What is the total vertical distance travelled by the ball between the first and sixth time the ball hits the ground?4. The sum of a geometric series with seven terms is 13,116, and the common ratio is r=3. Find the first term.5. A theater has 35 seats in the first row. Each row has four more seats than the row before it. Which expression represents the number of seats in the nth row?(1) 35+(n+4) (2) 35+(4n) (3) 35+(n+1)(4) (4) 35+(n-1)(4) ................
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