Unit 11: Sequences and Series

Transcription

1 Date Period Unit 11: Sequences and Series DAY TOPIC 1 Mathematical Patterns-Recursion 2 Arithmetic Sequences Arithmetic Means 3 Geometric Sequences Geometric Means 4 Arithmetic Series Summation Notation 5 6 Geometric Series Infinite Geometric Series 7 Mathematical Models 8 Review 1

2 Unit 11 (Sequences and Series) Day 1: Mathematical Patterns A 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 rebounds to 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: a 1 a2 a3... an 1 an an 1 A recursive formula defines the terms of a sequence by relating each term to the ones before it. Give your own example of a recursive sequence 4. 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 6 th and 7 th terms in the sequence. 2

3 b. Find the value of a 10 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. a1 a2 a3 a4 Length of a side Perimeter a. 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. The numbers in each row form a sequence a1 a2 a3 a4 a5 a6 length of a side perimeter a. For each sequence, find the next term (a7) and the twenty-fifth term (a25). b. Write the explicit formula for each sequence. c. Write the first 6 terms in the sequence showing the areas of the squares, then find a20. d. Write an explicit formula for the sequence from part (c) e. Given the recursive formula a n a n 1 3 can you find the 4 th term in the sequence? Explain. What is the difference between the two sequences, other than the fact that they are different numbers? 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

4 Unit 11 (Sequences and Series) Day 2: Arithmetic Sequences In 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, yes no common diff. b. 2, 5, 7, 12, yes no common diff. c. 48, 45, 42, 39 yes no common diff. d. 7, 10, 13, 16, yes no common diff. e. yes no cd. The formulas: Recursive explicit an a1 ( n 1) d 2. Suppose you participate in a bike-a-thon for charity. The charity starts with $1100 in donations. Each participant must raise at least $35 in pledges. What is the minimum amount of money raised if there are 75 participants? a. Why find the value of the 76 th term and not the 75 th term? 3. Use the explicit formula to find the 25 th 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 save at least $12 a week from the money you earn at a part-time job. In all, what is the minimum amount you will have after 26 weeks? 4

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. Graphs of arithmetic sequences are linear. Two terms of an arithmetic sequence and their arithmetic mean lie on the same line. 5. Find the missing term of the arithmetic sequence 84,, Find the missing term of the arithmetic sequence 24,, Write an expression for the arithmetic mean of a6 and a 7 8. Enter 2 arithmetic means: 9,,, 28.5 Closure: What is the difference between explicit and recursive? What are the arithmetic formulas for each? 5

6 Unit 11 (Sequences and Series) Day 3: Geometric Sequences and Geometric Means Geometric sequence- the ratio of any term to the previous term is constant The ratio between terms is called 1. Decide whether the sequence is geometric. a. 1, 2, 6, 24, 120, yes no common ratio b. 81, 27, 9, 3, 1,, yes no common ratio c. 5, 15, 45, 135, yes no common ratio d. 15, 30, 45, 60, yes no common ratio FORMULA Recursive formula explicit formula n 1 an a1 r 2. Write a rule for the nth term of the sequence -8, -12, -18, -27, then find a 8 3. Write a rule for the nth term then find a 8 : 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 the nth term. 5. One term of a geometric sequence is a4 3. The common ratio is r=3. Write the rule. 6. Find the 19 th term in each sequence a. 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 the product of the two numbers. 7. Find the missing term of each geometric sequence 6

7 a. 20,, 80, b. 3,, 18, 75, c. 28,, 5103, 8. Find the missing terms. 5,,, Find the missing terms: -2,,, 54 Closure: Can you state all the formulas discussed thus far? ARITHMETIC GEOMETRIC RECURSIVE EXPLICIT 7

8 Unit 11 (Sequences and Series) Day 4: Arithmetic Series and Summation Notation Arithmetic 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 by S n. n 2 Sum of an arithmetic series: sn a1 an Example: 1. Consider the following series: a. Find the sum of the first 30 terms a. find the sum of the first 25 terms 3. find the sum of the first 18 terms of Find the sum of the first 10 terms of: a b

9 Summation notation: Lower limit Upper limit 3 n 1 (5n 1) Explicit formula for the sequence 5. Use summation notation to write the series for 33 terms. 6. Use summation notation to write the series for 6 terms. 7. Use the series 3 (5n 1) to n 1 a. Find the number of terms in the series b. Find the first and last term of the series c. Evaluate the series 8. For each sum, find the number of terms, the first term, and the last term. Then evaluate the series. a. 10 ( n 3) b. n 1 5 n 2 n 2 Closure: List the explicit and recursive formulas learned thus far as well as the. 9

10 Unit 11 (Sequences and Series) Day 6: Geometric Series and Infinite Series Geometric series: the expression formed by adding the terms of a finite geometric sequence. Formula: S n n 1 r a 1 1 r Take a second and write all the formulas learned so far: Arithmetic sequence arithmetic series Geometric sequence geometric series Ex. 1. Consider the geometric series a. Find the sum of the first 10 terms 2. Find the sum of the first 8 terms of the geometric series Find the sum of the first 14 terms: In some cases you can evaluate an infinite geometric series. When r <1, the series converges, or gets closer and closer to the sum S. When r 1, the series diverges, or approaches no limit, therefore has no sum. 10

11 Decide whether each series has a sum (2) 3 9 n 1 n 1 The sum of an infinite geometric series with r <1 converges to the sum: S a1 1 r Evaluate each infinite geometric series

12 Unit 11 (Sequences and Series) Day 6: Geometric Series and Infinite Series- Extra Practice For each sum, find the number of terms, the first term, and the last term. Then evaluate the series n 3 4 ( n 1) 2. n 1 (3n 2) 6 (2n 1) 3. n 2 Use summation notation to write each arithmetic series for the specified number of terms ; n= ; n= ;n=6 Each sequence has 6 terms. Evaluate the related series 7. 1, 0, -1,, , 5, 6,, , -9, -11,, -17 RECAP Find the missing term of each geometric sequence ,, ,, 16 Write the explicit formula for each sequence. Then generate the first five terms. 12. a1 3, r a1 5, r a1 2, d 4 Find the 10 th term of the following , 14, 16, 18, 16. 3, 9, 27, 81, 17. 9, 5, 1, -3, 12

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

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15 Unit 11 (Sequences and Series) Day 8: U11 Review 1. Given the sequence, 2, 6, 10, 14.. a. Find the 100 th term. b. The term 7194 is in the sequence. Which term is it? c. Find S Insert 5 arithmetic means between 17 and 63. (Give your answers as fractions.) 3. Given the series 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. 4. Evaluate 5 2 n 5. Evaluate n n 1 2n 9 6. Given a geometric sequence with a first term of 100 and r = a. Find the 8 th term. b. Find the sum of the first 8 terms. 3, 4 7. Insert 2 geometric means between 38.9 and (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 100 th term is 512 and the common difference is 5. Find the first term. 15

16 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 2 nd year. a. How many inches will the tree grow during the 5 th 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 minute without 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 10 th 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 with 4.5% 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