Unit 11: Sequences And Series - Gradeamathhelp

DatePeriodUnit 11: Sequences and SeriesDAYTOPIC12Mathematical Patterns-RecursionArithmetic SequencesArithmetic Means3Geometric SequencesGeometric Means4Arithmetic SeriesSummation Notation56Geometric SeriesInfinite Geometric Series7Mathematical Models8Review1

Unit 11 (Sequences and Series) Day 1: Mathematical PatternsA sequence is an ordered list of numbers, where each number in a sequence is called a term.Examples:1. Start with a square with sides 1 unit long. On the right side, add on a square of the same size.Continue adding one square at a time in this way. Draw the first 4 figures of this pattern.a. Write the number of 1-unit segments in each figure above as a sequence.b. Predict the next term of the sequence.2. Suppose you drop a handball from a height of 10 feet. After the ball hits the floor, it reboundsto 85% of its previous height. How high will the ball rebound after its fourth bounce?a. after its seventh bounce?b. After what bounce will the rebound height be less than 2 feet?Notation: to represent terms of a sequence we often use variables such as:a1a 2 a3 . a n 1 a n a n 1A recursive formula defines the terms of a sequence by relating each term to the ones before it.Give your own example of a recursive sequence4. Describe the pattern that allows you to find the next term in the sequence 2, 6, 18, 54, 162, Write a recursive formula for the sequence.a. Find the 6th and 7th terms in the sequence.2

b. Find the value of a10 in the sequence.What is an explicit formula?5. The spreadsheet shows the perimeters of regular pentagons with sides from 1 to 4 units long.The numbers in each row form a sequence.a1a2a3a4Length of a side 1234Perimeter5101520a. For each sequence, find the next term (a 5 ) and the twentieth term (a 20 ).b. Write and explicit formula for each sequence.5. The spreadsheet shows the perimeters of squares with sides from 1 to 6 units long. Thenumbers in each row form a sequencea1 a2 a3 a4 a5 a6length of a side 1 2 3 4 5 6perimeter4 8 12 16 20 24a.b.c.d.e.For each sequence, find the next term (a7) and the twenty-fifth term (a25).Write the explicit formula for each sequence.Write the first 6 terms in the sequence showing the areas of the squares, then find a20.Write an explicit formula for the sequence from part (c)Given the recursive formula a n a n 1 3 can you find the 4th term in the sequence?Explain.What is the difference between the two sequences, other than the fact that they are differentnumbers? Can you predict what the next 3 numbers will be for each? Explain!!a. 1, 2, 3, 4, 5, b. 1,2,4,8,16, 3

Unit 11 (Sequences and Series) Day 2: Arithmetic SequencesIn an arithmetic sequence, the difference between two consecutive terms is constant.That difference is called the .1. Is the sequence arithmetic?a. 2, 4,8, 16, b. 2, 5, 7, 12, c. 48, 45, 42, 39 d. 7, 10, 13, 16, e.yesyesyesyesnonononocommon diff.common diff.common diff.common diff.yes noThe formulas:Recursivecd.explicitan a1 (n 1)d2. Suppose you participate in a bike-a-thon for charity. The charity starts with 1100 indonations. Each participant must raise at least 35 in pledges. What is the minimumamount of money raised if there are 75 participants?a. Why find the value of the 76th term and not the 75th term?3. Use the explicit formula to find the 25th term in the sequence 5, 11, 17, 23, 29, 4. Suppose you already saved 75 towards the purchase of a new iphone. You plan to saveat least 12 a week from the money you earn at a part-time job. In all, what is theminimum amount you will have after 26 weeks?4

The arithmetic mean of any two numbers is the average of two numbers.Some facts: For any three sequential terms in an arithmetic sequence, the middle term is thearithmetic mean of the first and third. Graphs of arithmetic sequences are linear. Two terms of an arithmetic sequence and their arithmetic mean lie on the sameline.5. Find the missing term of the arithmetic sequence 84, , 110.6. Find the missing term of the arithmetic sequence 24, , 577. Write an expression for the arithmetic mean of a6 and a78. Enter 2 arithmetic means: 9, , , 28.5Closure: What is the difference between explicit and recursive? What are the arithmeticformulas for each?5

Unit 11 (Sequences and Series) Day 3: Geometric Sequences and Geometric MeansGeometric sequence- the ratio of any term to the previous term is constantThe ratio between terms is called1. Decide whether the sequence is geometric.a. 1, 2, 6, 24, 120, yesb. 81, 27, 9, 3, 1,, yesc. 5, 15, 45, 135, yesd. 15, 30, 45, 60, yesFORMULARecursive formulanonononocommon ratiocommon ratiocommon ratiocommon ratioexplicit formulaan a1 r n 12. Write a rule for the nth term of the sequence -8, -12, -18, -27, then find a83. Write a rule for the nth term then find a8 : 5, 2, 0.8, 0.32, 4. One term of a geometric sequence is a3 5 . The common ratio is 2. Write a rule for thenth term.5. One term of a geometric sequence is a4 3 . The common ratio is r 3. Write the rule.6. Find the 19th term in each sequencea. 11, 33, 99, 297, b. 20, 17, 14, 11, 8, To find a geometric mean of any two positive numbers, take the positive square root of theproduct of the two numbers.7. Find the missing term of each geometric sequence6

a. 20, , 80, b. 3, , 18, 75, c. 28, , 5103, 8. Find the missing terms. 5, , , 1359. Find the missing terms: -2, , , 54Closure:Can you state all the formulas discussed thus far?RECURSIVEEXPLICITARITHMETICGEOMETRIC7

Unit 11 (Sequences and Series) Day 4: Arithmetic Series and Summation NotationArithmetic series: the expression formed by adding the terms of an arithmetic sequence.The sum of the first n terms of an arithmetic series is denoted bySum of an arithmetic series:sn Sn .n a1 an 2Example:1. Consider the following series: 4 7 10 13 16 19 a. Find the sum of the first 30 terms2. 20 18 16 14 a. find the sum of the first 25 terms3. find the sum of the first 18 terms of 100 120 140 4. Find the sum of the first 10 terms of:a. 2 6 10 14 18 b. 3 79 4 5 .228

Summation notation:Upper limit3 (5n 1)Explicit formula forthe sequencen 1Lower limit5. Use summation notation to write the series 3 6 9 for 33 terms.6. Use summation notation to write the series 1 2 3 for 6 terms.37. Use the series (5n 1) ton 1a. Find the number of terms in the seriesb. Find the first and last term of the seriesc. Evaluate the series8. For each sum, find the number of terms, the first term, and the last term. Then evaluatethe series.10a. (n 3)n 15b. n2n 2Closure: List the explicit and recursive formulas learned thus far as well as the.9

Unit 11 (Sequences and Series) Day 6: Geometric Series and Infinite SeriesGeometric series: the expression formed by adding the terms of a finite geometric sequence.Formula: 1 r nS n a1 1 r Take a second and write all the formulas learned so far:Arithmetic sequencearithmetic seriesGeometric sequencegeometric seriesEx.1. Consider the geometric series 1 5 25 125 625 a. Find the sum of the first 10 terms2. Find the sum of the first 8 terms of the geometric series 1 8 64 512 3. Find the sum of the first 14 terms: 1 4 16 64 In some cases you can evaluate an infinite geometric series. When r 1, the series converges, orgets closer and closer to the sum S. When r 1, the series diverges, or approaches no limit,therefore has no sum.10

Decide whether each series has a sum.1 1 .4. 1 5 255. 4 8 16 .1 16. 1 .3 97. 5(2)n 1n 1The sum of an infinite geometric series with r 1 converges to the sum:a1S 1 rEvaluate each infinite geometric series1 1 18. 1 .2 4 89. 3 3 3 3 .2 4 811

Unit 11 (Sequences and Series) Day 6: Geometric Series and Infinite Series- Extra PracticeFor each sum, find the number of terms, the first term, and the last term. Then evaluate theseries.41. (n 1)n 162. (2n 1)3.n 26 (3n 2)n 3Use summation notation to write each arithmetic series for the specified number of terms.4. 1 3 5 ; n 75. 4 8 12 ; n 46.10 7 4 ;n 6Each sequence has 6 terms. Evaluate the related series7. 1, 0, -1, , -48. 4, 5, 6, , 9 , -17RECAPFind the missing term of each geometric sequence.10. 4, , 169. -7, -9, -11,11. 9, , 16Write the explicit formula for each sequence. Then generate the first five terms.12. a1 3, r 213. a1 5, r 3a1 2, d 4Find the 10th term of the following.15. 12, 14, 16, 18, 16. 3, 9, 27, 81, 14.17. 9, 5, 1, -3,12

Unit 11 (Sequences and Series) Day 7: Word Problems (mathematical models)13

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Unit 11 (Sequences and Series) Day 8: U11 Review1. Given the sequence, 2, 6, 10, 14 .a. Find the 100th term.b. The term 7194 is in the sequence. Which term is it?c. Find S75 .2. Insert 5 arithmetic means between 17 and 63. (Give your answers as fractions.)3. Given the series 100 96 92.16 .a. Find the sum of the first 20 terms.b. For which value of n will the sum be approximately 1850?c. Find the sum of the infinite series.n2 n 2 35054. Evaluate5. Evaluate 2n 9n 136. Given a geometric sequence with a first term of 100 and r ,4a. Find the 8th term.b. Find the sum of the first 8 terms.7. Insert 2 geometric means between 38.9 and 98.6. (do not round r too much!)8. Given the sequence 5, 15, 45 , which term has the value 885,735?9. For an arithmetic sequence, the 100th term is 512 and the common difference is 5.Find the first term.15

10. Assume that the number of inches a tree grows up each year is a geometric sequence.Suppose the tree grows 40 inches the first year and 38 inches the 2nd year.a. How many inches will the tree grow during the 5th year?b. How tall will the tree be after a total of 10 years?c. Predict the ultimate height of the tree.11. Patty is learning to type. On the first day she types an average of 7 words per minutewithout making a mistake. Each day after that she types 2 additional words per minute.a. How many words per minute will she be able to type on the 10th day?b. She must be able to type 35 words per minute to interview for a data-entry job.How many days must she practice typing before she can interview for the job?12. A particular school board’s budget includes 50,000 for classroom technology this year,with an increase of 4% per school year.a. How much will the school district spend on classroom technology 6 years from now?b. What will be the total amount spent over the 6 years on classroom technology?13. Chris is saving money to buy a mountain bike. He puts 100 into a saving account with4.5% interest compounded monthly. (Remember to divide the rate by 12!)a. How much will be in the account after 6 months?b. If the bike costs 200, in how many months can he afford the bike?(He better make some deposits, because he’ll be waiting a long time!!)14. The 5 formulas I need to know are: (Try to write them without looking at your notes.)a. Arithmetic Sequence:b. Arithmetic Series:c. Geometric Sequence:d. Geometric Series:e. Infinite Geometric Series:16

Unit 11 (Sequences and Series) Day 2: Arithmetic Sequences . 5 The arithmetic mean of any two numbers is the average of two numbers. Some facts: For any three sequential terms in an arithmetic sequence, the middle term is the arithmetic mean of the first and third.