COLLEGE ALGEBRA. Arithmetic & Geometric Sequences

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1 COLLEGE ALGEBRA By: Sister Mary Rebekah Cornell-Style Fill in the Blank Notes and Teacher s Key Arithmetic & Geometric Sequences 1

2 Topic: Discrete Functions main ideas & questions What is a DISCRETE Function? notes & examples Discrete VERSUS Continuous Directions: Indicate whether the following functions are discrete or continuous. 1. 1,1, 0,0, (1,1) 2. yy = 1 3 xx Mary goes to the local plant nursery to buy some lavender seeds. For every packet she buys, she gets 25 seeds yy = xx 3 + 3xx 8. As the date get closer to July 20, the temperature at 3:00 increases. 9. Gracen gets one $5 coupon for every $200 he spends at an online store. 2

3 Topic: Arithmetic Sequences- Lab Activity Example A company produces rods by joining cubes together from end to end. It also places one sticker upon each exposed face of each cube on the rods. Every exposed face of each cube has to have exactly one sticker; this rod of length 2, would then need 10 stickers. 1. How many stickers would you need for rods of length 1-10? #Cubes # S tickers 2. How many stickers would you need for a rod of length 20? 56? Please, explain how you determine these values. 3. How many stickers would you need for a rod of length 137? 213? Please, explain how you determined these values. 4. Write a general rule that would allow you to find the number of stickers needed for a rod of any length. Explain your rule. 3

4 Topic: Intro to Sequences main ideas & questions WHAT IS A Sequence? FINITE Sequence INFINITE Sequence TERM NOTATION Sequences AS FUNCTIONS notes & examples Sequence: Term: example: example: The first term in a sequence is denoted by. (Read a sub one) Each subsequent term is denoted by, ( a sub n), where n is. Example: Given the following sequence 1, 5, 9, 13, 17, Identify the following term values: aa 11 : aa 22 : aa 44 : aa 99 : aa 1111 : Sequences can be considered a type of discrete function. Since each term is paired with exactly one term number, a sequences is a function with the following properties. The domain is the set of. nn aa nn The range is the set of. Recursive FORMULAS A rule for sequences by which the in the sequence is defined by the. In other words, one (or more) of the previous terms are used to generate the next term. Directions: Find the first 5 terms in the following sequences. Examples Example #1 aa 1 = 14, aa nn = aa nn Example #2 aa 1 = 6, aa nn = 2aa nn 1 5 4

5 Example #3 aa 1 = 354, aa nn = aa nn 1+6 (for n >2) 3 Example #4 aa 1 = 3, aa 2 = 4; aa nn = aa nn 1 aa nn 2 (for n >3) Directions: Write a recursive formula (rule) for the following sequences. Then give the next three terms. Example #5 4, 11, 32, 95, 284, Example #6 100, 60, 40, 30, 25, Explicit FORMULAS Examples Directions: Find the first five terms of each sequence. 7. aa nn = 7(nn 3) 8. aa nn = 1 3 nn+1 9. aa nn = 4nn aa nn = 2nn(nn + 5) 11. aa nn = 2 nn aa nn = nn 4 2 Directions: Write an explicit rule for ach sequence; then find the next three terms. 13. { 7, 9, 11, 13, 15, } 14. {3, 6, 11, 18, 27, } 15. {2, 2 / 5, 3, 7 / 2, 4, } 16. 1, 2, 3, 2, 5, 5

6 Topic: Intro to Series & Summations main ideas & questions notes & examples WHAT IS A series? Series: sequence {1, 2, 3, 4} {3, 6, 9, 12, } 1 2, 1 4, 1 8, 1 16, series PARTIAL Sums Partial sum: Directions: Find the partial sum for each given sequence. 1. 1, 2, 3, 4, 5, ; find SS , 7, 13, 25, 49, ; find SS , 4, 9, 16, 25, ; find SS ,, 1, 1, 1, ; find SS SUMMATION notation A way to represent a series using the Greek letter Σ, Sigma, to denote the sum. 5 2nn nn=1 Find the sum of the series above. EXAMPLES Directions: Find the sum of each series (nn 1) nn= ( 3nn) nn=1 6

7 7. 4 (5kk + 2) kk= (2aa 7) aa= (yy 2 + yy) yy= nn=2 nn (mm 2 1) mm= pp 2 pp= (xx 2 5) xx=2 7

8 Topic: Arithmetic Sequences & Series main ideas & questions WHAT IS AN arithmetic SEQUENCE? COMMON difference notes & examples Arithmetic sequence: Common difference: Directions: Determine whether the sequence is arithmetic. If yes, identify the common difference and give the next three terms. 1. 5, 8, 11, 14, 17, 2. 23, 16, 9, 2, 5, 3. 1, 2, 3, 5, 8, 4. 6, 8, 10 12, 14, 5. 20, 10, 5, 0, 5, 6. 35, 23, 11, 1, 13, EXPLICIT Formula FOR ARITHMETIC SEQUENCES EXAMPLES The n th term of any arithmetic sequence can be found using the formula: Where aa 1 is the and dd is the. Directions: Write a rule for each sequence, then find the indicated term. 7. 5, 1, 3, 7, ; aa , 11, 2, 8, 17, ; aa , 9, 12, 15, ; aa , 9.5, 11, 12.5, ; aa 48 8

9 Directions: If the given terms are part of an arithmetic sequences, write a rule for the sequences, then find aa aa 1 = 3 and aa 3 = aa 1 = 17 and aa 24 = 6 ARITHMETIC Series Recall that a series is the of the terms in a sequence. When finding the sum of an arithmetic series, use the following formula. Where nn is the. aa 11 is the, and aa nn is the. Examples Directions: Find the indicated sum for each arithmetic series ; ss ; ss (6xx 4) ( 2xx + 3) xx=2 xx=8 9

10 Topic: Arithmetic Sequences & Series Application Examples OF ARITHMETIC SEQUENCES & SERIES 17. Sequence For tax purposes, a firm depreciates its $900,000 building over 30 years by the straight-line method, which depreciated the value of a building by 900,000 = 30,000 dollars each year. Write a sequence that gives 30 the value of the building at the end of each of the first five years. Write out how you got this sequence. 18. Sequence Accruing interest on a principle. Month Interest, I =Prt ($) Future value of the investment 1 (5000)(0.01)(1)= = 2 (5000)(0.01)(1)= = 3 (5000)(0.01)(1)= = 4 (5000)(0.01)(1)= = 5 (5000)(0.01)(1)= = 6 (5000)(0.01)(1)= = 19. Series Football Contracts. Suppose that a football player is offered a chance to sign a contract for 18 games with the following salary plan: Plan: $10,000 for the first game with a $10, 000 raise for each game thereafter. 10

11 Topic: Geometric Sequences main ideas & questions WHAT IS A geometric SEQUENCE? COMMON ratio notes & examples geometric sequence: Common ratio: Directions: Determine whether the sequence is geometric. If yes, identify the common ratio and give the next three terms. 1. 6, 12, 24, 48, , 270, 90, 30, 3. 2, 10, 50, 250, 4. 4, 8, 20, 60, 5. 4, 1, 1 4, 1 16, 6. 2, 4, 16, 256, 7. 2, 14, 98, 686, , 8, 20, 50, EXPLICIT Formula FOR GEOMETRIC SEQUENCES The n th term of any geometric sequence can be found using the formula: Where aa 1 is the and rr is the. EXAMPLES Directions: Write a rule for each sequence, then find aa , 5, 25, 125, , 65, 62.5, 16.25, 11

12 Examples 11. 2, 8, 32, 128, 12. 8, 24, 72, 216, , 90, 60, 40, , 800, 200, 50, Directions: Use the information given to find the indicated value. 15. aa 1 = 7, rr = 4; find aa aa 5 = 68, rr = 0.5; find aa aa 1 = 13, aa 4 = 4459; find aa aa 1 = 6, aa 5 = 0.048; find aa 4 Application 19. Jackson bought a car for $38,000. If the car depreciates in value by 20% each year, what will the car be worth in 8 years? 12

13 Topic: Geometric Sequence Application main ideas & questions WHAT IS A geometric SEQUENCE, AGAIN? CONSIDER THIS problem notes & examples geometric sequence: Football contacts: Football Contracts. Suppose that a football player is offered a chance to sign a contract for 18 games with one of the following salary plans: Plan 1: $10,000 for the first game with a $10, 000 raise for each game thereafter. Plan 2: $7 for the first game with his salary double for each game thereafter. Which salary plan should he accept if he wants the make the most money on his last game? Lets breakdown the pattern os a geometric sequence. Examining plan 2 of the football contract, the player gets $7 for the first game, and his salary doubles for each game thereafter. Game (n) Amount In other words a sub what? EXPLICIT Formula FOR GEOMETRIC SEQUENCES The n th term of any geometric sequence can be found using the formula: Where aa 1 is the and rr is the. 13

14 Topic: Compound Interest is a Geometric Sequence main ideas & questions COMPOUND Interest notes & examples The addition of interest to the principal sum of a loan or deposit, or in other words, interest on interest. It is the result of reinvesting interest, rather than paying it out, so that interest in the next period is then earned on the principal sum plus previously accumulated interest. P= r= t= n= Examples 1. Your 3 year investment of $20,000 received 5.2% interested compounded annually. What is your total return? 2. You borrowed $59,000 for 2 years at 11% which was compounded annually. What total will you pay back? 3. Your allowance of $190 got 11% compounded monthly for 1 2/3 years. What s it worth after the 1 2/3 years? 4. Your $440 gets 5.8% compounded annually for 8 years. What will your $440. be worth in 8 years? 14

15 Topic: Geometric Series main ideas & questions What is a GEOMETRIC series? Examples notes & examples A geometric series is the of a geometric sequences. To find the sum, use the following formula: Where n is the, aa 1 is the, and rr is the. *This is for FINITE geometric series! Directions: Find the indicated sum for each FINITE geometric series ; ss ; ss nn nn=1 8 nn 1 nn=1 15

16 Infinite GEOMETRIC series Convergent SERIES formula Examples 16

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