Algebra 2/Trig: Chapter 6 Sequences and Series In this unit, we will

Transcription

1 Algebra 2/Trig: Chapter 6 Sequences and Series In this unit, we will Identify an arithmetic or geometric sequence and find the formula for its nth term Determine the common difference in an arithmetic sequence Determine the common ratio in a geometric sequence Determine a specified term of an arithmetic or geometric sequence Specify terms of a sequence, given its recursive definition Represent the sum of a series, using sigma notation Determine the sum of the first n terms of an arithmetic or geometric series

2 Table of Contents Day : Arithmetic and Geometric Sequences SWBAT: Identify an arithmetic or geometric sequence and find the formula for its nth term Determine the common difference in an arithmetic sequence Determine the common ratio in a geometric sequence Determine a specified term of an arithmetic or geometric sequence Specify terms of a sequence, given its recursive definition Pgs. 6 in Packet HW: Pgs. 7 9 in Packet Day 2: More with Arithmetic & Geometric Sequences SWBAT: Identify an arithmetic or geometric sequence and find the formula for its nth term Determine the common difference in an arithmetic sequence Determine the common ratio in a geometric sequence Determine a specified term of an arithmetic or geometric sequence Specify terms of a sequence, given its recursive definition Pgs. 0 4 in Packet HW: Pgs. 5 7 in Packet QUIZ on Day 3 ~ 7 min Day 3: Series SWBAT: Represent the sum of a series, using sigma notation Determine the sum of the first n terms of an arithmetic or geometric series Pgs in Packet HW: Pgs in Packet Day 4: Sequences and Series Mixed Practice SWBAT: Review problems involving Sequences and Series Pgs in Packet QUIZ on Day 4 ~ 5 min Day 5: Practice Test (Not in Packet) Day 6: Test

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4 Lesson #: Arithmetic and Geometric Sequences Definition of a sequence: Example : Example 2: Concept : Ways to define a sequence There are two ways to define a sequence: or. An explicitly defined sequence is like a formula. Plugging into the formula gives the terms of the sequence. Subscripts name terms. They are not values in the problem. Example 3: Consider sequence. = 3n + 2. Find the first 3 terms ( ) of this A recursively defined sequence has two parts; () It gives the first term and (2) all of the other terms are found using operations on the previous term(s) Key Points for Recursive Formulas Subscripts name terms. They are not values in the problem. If the next term is For example, if an the term before it will be an because n- an is a3, an is a2. If the next term is For example, if an the term before it will be an because n an is a3, an is a2. Bottom Line: Build off the last term! is one smaller than n. is one smaller than n+.

5 2 For each problem, find the next four terms. Example 4: n n a a a Example 5: 5 n n a a a n Label the following as either an Explicit Formula or a Recursive Formula.

6 Arithmetic Sequences If a sequence of values follows a pattern of adding a fixed amount from one term to the next, it is referred to as an arithmetic sequence. The number added to each term is constant (always the same). The fixed amount is called the common difference, d. a) The following is an example of an arithmetic sequence: 3,8,3,8,23... What is the common difference, d? 7 b) What is the common difference of the following arithmetic sequence?: 5,, 2,, Geometric Sequences If a sequence of values follows a pattern of multiplying a fixed amount (not zero) times each term to arrive at the following term, it is referred to as a geometric sequence. The number multiplied each time is constant (always the same). The fixed amount is called the common ratio, r. a) The following is an example of a geometric sequence: 8,56,392, What is the common ratio, r? b) What is the common ratio of the following geometric sequence?: 27, 9, 3,,,... Concept 2: Generating a Sequence A sequence can be defined by a formula (or generator) which generates each term. (Note: the variable n appears in most generator. It is used to indicate the position of a term in a sequence.) The formula to generate any arithmetic sequence can be written in the form: The formula to generate any geometric sequence can be written in the form: 3

7 Example 6: Find a formula to generate the arithmetic sequence 3, 5, 7, and use it to generate the 50th term. Step : Step 2: Example 7: Find a formula to generate the geometric sequence 4, 2, 36, and use it to determine the 9th term. Step : Step 2: 4

8 You Try it! Determine if the sequence is arithmetic. If it is, find the common difference, the term named in the problem, and the explicit formula. Determine if the sequence is geometric. If it is, find the common ratio, the term named in the problem, and the explicit formula. 5

9 SUMMARY Exit Ticket. 2. 6

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13 Lesson #2: More with Arithmetic & Geometric Sequences WARM-UP! Writing Sequences using equations practice ) a n 3n 2 What type of sequence is this? Write the first 5 terms of the sequence: 2) 62 3 n a n What type of sequence is this? Write the first 4 terms of the sequence: 3) a ) n n 3( 2 What type of sequence is this? Write the first 5 terms of the sequence: 2 a 6 ( n ) 4) n What type of sequence is this? Write the first 5 terms of the sequence: 5) a and the common difference of this arithmetic sequence is -3. Write the first 4 terms of the sequence: Write the explicit equation of this sequence: 6) a 2 and the common ratio of this geometric sequence is 0. Write the first 4 terms of the sequence: Write the explicit equation of this sequence: 0

14 , 3, 2,,,... 7) Write an explicit equation for the sequence: ) Write an explicit equation for the sequence: -7.8, -5.6, -3.4, -.2,...

15 Concept : Arithmetic Mean Arithmetic mean- the mean average between any two numbers of a sequence -a missing term can be found by finding the arithmetic mean of two terms. Ex: Given the arithmetic sequence 84,, 0, find the missing term. Arithmetic mean = = = 97 Ex: Find the missing term of each arithmetic sequence. ) 6,, 36 2) 23,,, 7 3) 2,,,, 0 4) If = 80 and = 32 in a arithmetic sequence, find the 24 th term. 2

16 Concept 2: Geometric Mean Geometric mean- the positive square root of the product of two numbers of a sequence - A missing term can be found by finding the geometric mean of two terms Ex: Given: 20,, 80 Step : Step 2: Step 3: Find the missing term of each geometric sequence. ) 3,, ) 28,,, ) 9,683,,,, 243 4) If = -6 and = -296 in a geometric sequence, find the 4th term. 3

17 SUMMARY Exit Ticket ) 2) 4

18 HW Day 2 Two important types of sequences are arithmetic sequences and geometric sequences. Try to figure out which is which, and fill in the missing number in each sequence. arithmetic or geometric ) 4, 7, 0,, 6,... 2) 2, 4, 8, 6,,... 3),, 9, 27, 8,... 4) 3.5, 6, 8.5,,,... 5) 8, 2, 8,, 40.5,... 6), -5.5, -9.5, -3.5,... 7) 256, 64, 6, 4,,... 8) -4, 8, -6, 32, -64,,... 5

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21 Day 3 Series Review of Summation Notation Last value for the index variable Symbol that tells us to add Index variable (the one we plug the values into) 5 k 2 k Plug the values into this expression and add up each term First value for the index variable Find each sum a a 2a 2. A series is the sum of the terms in a sequence. On the first page of this lesson, you reviewed the different ways we have used summation notation so far this year.. Which of the following represents the sum ? a (2) () a0 a 8 4( a ) (3) 5 a0 8 4a (4) 4 a0 8 4a Solution: 8

22 You try it! 2. Which of the following represents the sum ? ( n ) () (2) n n n2 (3) 9 2 n (4) n 80 n3 n 3. Imagine having to find the sum of the arithmetic series by hand. Imagine having to find the sum of the geometric series 25 k 4(3) k by hand. Luckily there are formulas to find the sum of the first n terms of any arithmetic or geometric series. Even more luckily, you do not have to memorize them because they are given to you on the A2&T reference sheet. The rest of this lesson will deal with some problems you might encounter where you need to use these formulas. You will also need to use the other formulas from the previous lessons to find the numbers to plug into the formulas. Finding the nth term in an arithmetic sequence: 9

23 Finding the nth term in a geometric sequence:. Find the sum of the first 28 terms of the series (Hint: You need to find a28 first). 2. Find the sum of the first 2 terms of the series, (Since this one is geometric, we do not need a 2 ) 3. Evaluate this series using the series formula: 4. Evaluate using the series formula: 20

24 Find the sum of the first 20 terms of the sequence 4, 6, 8, 0,... 2

25 9. Find the sum of the sequence -8, -5, -2,..., 7 0. Find the sum of the first 8 terms of the sequence -5, 5, -45, 35,... SUMMARY Exit Ticket 22

26 Day 3 Series HW Sum of Arithmetic Series Name Sum of a Finite Arithmetic Series: S n n( a a n ) 2 ) Evaluate each arithmetic series. a) a 4, an 22, n 0 b) 2, a 56, n 0 a n 00 c) (2n ) d) (3n 2) n 50 n e) 20, 23, 26, 29,... n=25 f) 20, 30, 40, 50,... n=30 23

27 2) Determine the number of terms n in the following arithmetic series. a) a 9, a n 96, S 690 b) a 5, a n 79, S 423 n n 3) Determine the sum of the all of the even integers from 2 to ) A company offers Sue a starting salary of $50,000 plus a guaranteed pay increase of 5,000 each year. What is the total amount of money Sue will have earned after working for 25 years? 24

28 Sum of Geometric Series Name Sum of a Finite Geometric Series: S n a ( r r n ) ) Evaluate each geometric series. 7 a) (4 k k ) 2 b) 43 n n c), -4, 6, -64,... n=9 d), 2, 4, 8,... n=30 e) a 4, an 8748, r 3 f) 4, a 024, r 2 a n 25

29 2) Determine the number of terms n in the following geometric series. a) a 2, r 5, S 62 b) a 3, r 3, S 60 n n 3) A company offers Jim a starting salary of $50,000 plus a guaranteed pay increase of 5% each year. What is the total amount of money Jim will have earned after working for 25 years? 26

30 Answers to Arithmetic Series Answers to Geometric Series 27

31 Day 4 - Sequences and Series Mixed Practice 204. What is the difference between an arithmetic and a geometric sequence? 2. Find the next three terms of each sequence a. 9, 6, 23,,, b. 00, -200, 400,,, c. -8, -5, -2,,, 3. Find the first three terms of each sequence where d is the common difference, and r is the common ratio a. a 576, r,, 2 b. a 2, d 3,, c. 5 3 a, d,, Find a 8 if a n 43n. 5. Find a 7 if 2 2 n a. n 6. Find a 2 for -7, -3, -9, 7. Find a 8 for 4, -2, 36, 28

32 8. Find 4 a if a 3 and the common difference is d 7 9. Find 8 a if the common ratio r 3 and a 3 0. Write the equation for the nth term for each sequence a. 7, 6, 25, 34, b. 36, 2, 4,. Find the 0 th term of the arithmetic sequence given the following information: a, a Find the 7 th term of the geometric sequence given the following information: a, a Write the first 5 terms of each sequence described below with recursive equations: 3. a 3 a n 3a n 0 4. a 5 a n a n 4 5. a a a 2 2 n2 a n 4a n 29

33 Series Questions Find the value of each series below (2 j 6) j 3 7. k0 2(4) k j j j 6 9. Find 2 x k if x 2, x 4, x 8 and x 0 k Write this series using sigma notation: Write this series using sigma notation:

34 Evaluate each arithmetic series. 22. a, a 33, S? (7n ) n Evaluate each geometric series n n 25., 2, 4, 8,... n = Determine the number of terms n in the following geometric series. a 2, r 5, Sn 62 3

35 27) Expand the following binomials. (Remember to fully simplify each term.) a) 5 ( x 3) b) 6 ( a 2) c) ( 3x 2y) 3 d) Find the 3 rd term of ( 2 y 2 7 a 3 ). 32