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# Sequences and series

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### Sequences and series

1. 1. Sequences …and Series Exploring Arithmetic and Geometric Sequences
2. 2. Sequences <ul><li>1, 2, 3, 4, 5, 6, 7, … </li></ul><ul><li>1, 3, 5, 7, 9, 11, … </li></ul><ul><li>2, 4, 8, 16, 32, 64, … </li></ul><ul><li>1, 1, 2, 3, 5, 8, 13, 21, … </li></ul><ul><li>27, 20, 13, 6, -1, -8, … </li></ul><ul><li>40, 20, 10, 5, 2.5, 1.25, … </li></ul><ul><li>3, 9, 27, 81, 243, 729, … </li></ul>
3. 3. Arithmetic Sequences <ul><li>Ex.1) 2, 5, 8, 11, 14, 17, 20, … </li></ul><ul><li>Ex. 2) 4, 9, 14, 19, 24, 29, 34, … </li></ul><ul><li>Ex. 3) 1, 8, 15, 22, 29, 36, … </li></ul><ul><li>Notes: </li></ul><ul><ul><li>Lists of Numbers </li></ul></ul><ul><ul><li>Pattern </li></ul></ul><ul><ul><li>Set difference between terms </li></ul></ul>
4. 4. Generalizing Arithmetic Sequences <ul><li>Ex.1) 2, 5, 8, 11, 14, 17, 20, … </li></ul><ul><ul><li>5 – 2 = 3, 8 – 5 = 3, 11 – 8 = 3 … 20 – 17 = 3 </li></ul></ul><ul><ul><li>Difference = 3 = d </li></ul></ul><ul><ul><li>(2 + 3) = u 2 , (5 + 3) = u 3 , (u 2 + 3) = u 3 </li></ul></ul><ul><ul><li>(2 + 3 + 3) = u 3 , 2 + 2(3) = u 3 , 2 = 3 – 1 </li></ul></ul><ul><ul><li>u n = (2 + 3(n – 1)) </li></ul></ul><ul><li>Ex. 3) 1, 8, 15, 22, 29, 36, … </li></ul><ul><ul><li>8 – 1 = 7, 15 – 8 = 7, 22 – 15 = 7 … 36 – 29 = 7 </li></ul></ul><ul><ul><li>d = 7 </li></ul></ul><ul><ul><li>(1 + 7) = u 2 , (8 + 7) = u 3 </li></ul></ul><ul><ul><li>(1 + 7 + 7) = u 3 , 1 + 2(7) = u 3 , 2 = 3 – 1 </li></ul></ul><ul><ul><li>u n = (1 + 7(n – 1)) </li></ul></ul><ul><li>Notes: </li></ul><ul><ul><li>Determine common difference </li></ul></ul><ul><ul><li>General formula: u n = u 1 + (n – 1)d </li></ul></ul>
5. 5. Geometric Sequences <ul><li>Ex. 1) 2, 4, 8, 16, 32, 64, … </li></ul><ul><li>Ex. 2) 1000,100, 10, 1, 0.1, 0.01, … </li></ul><ul><li>Ex.3) 3, 9, 27, 81, 243, 729, … </li></ul><ul><li>Notes: </li></ul><ul><ul><li>List of numbers </li></ul></ul><ul><ul><li>Pattern [Multiplication] </li></ul></ul><ul><ul><li>Common ratio [r] </li></ul></ul>
6. 6. Generalizing Geometric Sequences <ul><li>Ex. 1) 2, 4, 8, 16, 32, 64, … </li></ul><ul><ul><li>4÷2 = 2, 8÷4 = 2, … 64÷32 = 2 </li></ul></ul><ul><ul><li>Common ratio r = 2 </li></ul></ul><ul><ul><li>u 1 ×2 = u 2 , u 2 ×2 = u 3 , (u 1 ×2)×2 = u 3 </li></ul></ul><ul><ul><li>u 3 = u 1 ×(2 2 ) , thus u n = u 1 (2 n-1 ) </li></ul></ul><ul><li>Ex. 2) 1000,100, 10, 1, 0.1, 0.01, … </li></ul><ul><ul><li>100÷1000 = 1 / 10 , 10÷100 = 1 / 10 </li></ul></ul><ul><ul><li>Common ratio r = 1 / 10 </li></ul></ul><ul><ul><li>1 / 10 × u 1 = u 2 , 1 / 10 × u 2 = u 3 , 1 / 10 × ( 1 / 10 × u 1 )= u 3 </li></ul></ul><ul><ul><li>u 3 = u 1 ×( 1 / 10 ) 2 , thus u n = u 1 ( 1 / 10 ) n-1 </li></ul></ul><ul><li>Notes: </li></ul><ul><ul><li>Find common ration r [dividing one term by previous term] </li></ul></ul><ul><ul><li>General formula: u n = u 1 (r n-1 ) </li></ul></ul>
7. 7. Series <ul><li>Arithmetic Series </li></ul><ul><ul><li>2 + 5 + 8 + 11+ 14 +17 + 20 +… </li></ul></ul><ul><ul><li>4 + 9 +14 +19 + 24 + 29 + 34 +… </li></ul></ul><ul><li>Geometric Series </li></ul><ul><ul><li>2 + 4 + 8 + 16 + 32 + 64… </li></ul></ul><ul><ul><li>3 + 9 + 27 + 81 + 243 + 729… </li></ul></ul>What is a series? <ul><ul><li>Lists of numbers with patterns </li></ul></ul><ul><ul><li>Terms of sequences added together </li></ul></ul>
8. 8. Applications <ul><li>What are Arithmetic Sequences used for? </li></ul><ul><ul><li>Solving Math Problems </li></ul></ul><ul><ul><li>Used in conjunction with Series [Arithmetic Series] </li></ul></ul><ul><ul><ul><li>Can be used to sum up long lists of numbers </li></ul></ul></ul><ul><ul><ul><li>Can be used to sum up prices </li></ul></ul></ul><ul><ul><ul><li>Ex. Theater tickets are sold for \$60 for odd numbered tickets and \$40 for even numbered tickets. There are 50 seats in the theater, how much money do even tickets bring in if all are sold? </li></ul></ul></ul><ul><li>What are Geometric Sequences used for? </li></ul><ul><ul><li>Solving Compound Interest </li></ul></ul><ul><ul><ul><li>When money is put into a bank it gains a percentage interest. </li></ul></ul></ul><ul><ul><ul><li>At any given point in time geometric series can determine how much money is in the bank, with interest. </li></ul></ul></ul><ul><ul><ul><li>To calculate population growth, often in biology. </li></ul></ul></ul>
9. 9. Recognizing Sequences <ul><li>1, 2, 3, 4, 5, 6, 7, … </li></ul><ul><li>1, 3, 5, 7, 9, 11, … </li></ul><ul><li>2, 4, 8, 16, 32, 64, … </li></ul><ul><li>1, 1, 2, 3, 5, 8, 13, 21, … </li></ul><ul><li>27, 20, 13, 6, -1, -8, … </li></ul><ul><li>40, 20, 10, 5, 2.5, 1.25, … </li></ul><ul><li>3, 9, 27, 81, 243, 729, … </li></ul>
10. 10. Properties <ul><li>Arithmetic Sequences </li></ul><ul><li>Geometric Sequences </li></ul><ul><li>A list of numbers </li></ul><ul><li>There is a pattern and order </li></ul><ul><li>Get one term by adding a number to the previous term in the sequence </li></ul><ul><li>There is a common difference d </li></ul><ul><li>n represents position in the list (nth term) </li></ul><ul><li>General formula: u n =u 1 +(n-1)d </li></ul><ul><li>Series adds each term in a sequence together </li></ul><ul><li>Can be used to sum lists of numbers, cost and money </li></ul><ul><li>A list of numbers </li></ul><ul><li>There is a pattern and order </li></ul><ul><li>Get one term by multiplying the previous term by a set number </li></ul><ul><li>There is a common ratio r </li></ul><ul><li>n represents position in the list (nth term) </li></ul><ul><li>General formula: u n = u 1 r n-1 </li></ul><ul><li>Series adds each term n a sequence together </li></ul><ul><li>Can be used to determine growth, population growth and interest </li></ul>
11. 11. Properties - Similarities <ul><li>Arithmetic Sequences </li></ul><ul><li>Geometric Sequences </li></ul><ul><li>A list of numbers </li></ul><ul><li>There is a pattern and order </li></ul><ul><li>Get one term by adding a number to the previous term in the sequence </li></ul><ul><li>There is a common difference d </li></ul><ul><li>n represents position in the list (nth term) </li></ul><ul><li>General formula: u n =u 1 +(n-1)d </li></ul><ul><li>Series adds each term in a sequence together </li></ul><ul><li>Can be used to sum lists of numbers, cost and money </li></ul><ul><li>A list of numbers </li></ul><ul><li>There is a pattern and order </li></ul><ul><li>Get one term by multiplying the previous term by a set number </li></ul><ul><li>There is a common ratio r </li></ul><ul><li>n represents position in the list (nth term) </li></ul><ul><li>General formula: u n = u 1 r n-1 </li></ul><ul><li>Series adds each term n a sequence together </li></ul><ul><li>Can be used to determine growth, population growth and interest </li></ul>
12. 12. Properties - Differences <ul><li>Arithmetic Sequences </li></ul><ul><li>Geometric Sequences </li></ul><ul><li>A list of numbers </li></ul><ul><li>There is a pattern and order </li></ul><ul><li>Get one term by adding a number to the previous term in the sequence </li></ul><ul><li>A common difference d </li></ul><ul><li>n represents position in the list (nth term) </li></ul><ul><li>Gen. formula: u n =u 1 +(n-1)d </li></ul><ul><li>Series adds each term in a sequence together </li></ul><ul><li>Can be used to sum lists of numbers, cost and money </li></ul><ul><li>A list of numbers </li></ul><ul><li>There is a pattern and order </li></ul><ul><li>Get one term by multiplying the previous term by a set number </li></ul><ul><li>There is a common ratio r </li></ul><ul><li>n represents position in the list (nth term) </li></ul><ul><li>General formula: u n = u 1 r n-1 </li></ul><ul><li>Series adds each term n a sequence together </li></ul><ul><li>Can be used to determine growth, population growth and interest </li></ul>
13. 13. Compare and Contrast <ul><li>Lists of numbers </li></ul><ul><li>Pattern and order </li></ul><ul><li>n is the position in the sequence </li></ul><ul><li>series adds the terms in the sequence together </li></ul><ul><li>has common difference d </li></ul><ul><li>get one term by adding or subtracting a set number to the previous term </li></ul><ul><li>general formula: </li></ul><ul><li>u n = u 1 + (n – 1)d </li></ul><ul><li>Can be used to find sums of long lists of numbers </li></ul><ul><li>Has a common ratio r </li></ul><ul><li>Get one term by multiplying the previous term by a set number </li></ul><ul><li>General formula: </li></ul><ul><li>u n = u 1 r n-1 </li></ul><ul><li>Can be used to figure out compound interest and population growth </li></ul>
14. 14. Now for Review <ul><li>Get into groups of four </li></ul><ul><li>Review information presented in class </li></ul><ul><li>Write helpful information on index cards [keep it on the lines – one side] </li></ul><ul><li>5 minutes </li></ul>