Arithmetic and Geometric Sequences Review

Transcription

1 Name Date 6-8 Sequences Word Problems Arithmetic and Geometric Sequences Review ARITHMETIC SEQUENCE GEOMETRIC SEQUENCE ( ) How do you know when to use the arithmetic sequence formula? How do you know when to use the geometric sequence formula? How do you find the common difference of two consecutive terms of an arithmetic sequence? How do you find the common ratio of two consecutive terms of a geometric sequence? 326, 321, 316, 311, 486, 162, 54, How do you find the common difference of two non-consecutive terms of an arithmetic sequence? U1 = 145, U5 = 113 How do you find the common ratio of two non-consecutive terms of an geometric sequence? U2 = 9, U4 = 36

2 Arithmetic and Geometric Sequence Word Problems How do you determine if a word problem is referring to an arithmetic sequence or a geometric sequence? 1. Look for keywords or clues in the question that tell you if the sequence is arithmetic or geometric. Arithmetic Sequence arithmetic, common difference, increases by a #, decreases by a #, etc.. Geometric Sequence geometric, common ratio, increases by a %, decreases by a %, etc.. 2. If they give you consecutive numbers (terms) try to see if there is a pattern to get from one term to the next Arithmetic Sequence adding or subtracting by a fixed number to get the next term Geometric Sequence multiplying or dividing by a fixed number to get the next term 1. The number of apartments in a housing development has been increasing by a constant amount every year. At the end of the first year the number of apartments was 150, and at the end of the sixth year the number of apartments was 600. (a) Calculate the number of apartments that the housing development increases by each year. (b) Calculate the number of apartments in the housing development at the end of the tenth year. (c) Calculate the number of years it will take for there to be 2040 apartments in the housing development.

3 2. On Monday Paco goes to a running track to train. He runs the first lap of the track in 120 seconds. Each lap he runs takes 1.06 times as long as his previous lap. (a) Calculate the number of seconds it will take to Paco to run the fourth lap. (b) Calculate the total amount of seconds it will take to Paco to run the first four laps. Geometric Sequence Word Problems with Percentages If a question refers to a percent, this means you are dealing with a geometric sequence. When given a percent, the common ratio is the percent remaining of the previous term. DECREASE (LOSS) INCREASE (GAIN) Ex: The current depression has caused a company to lose 12% of its current revenue each month during the year The company started out with $52,000 in the first month. If this pattern continues, how much money, to the nearest dollar, would the company have left after 8 months? Ex: Ben is playing a game similar to Monopoly. Each time Ben passes GO he receives 8% of the amount he already has. Ben starts with $100. How much money will Ben have after he passes GO 10 times?

4 Practice: Identify the common ratio in each situation. a. A certain water filtration system can remove 70% of the contaminants each time a sample of water is passed through it. b. John s salary earns an increase of 4% each successive year. 3. On Vera s 18 th birthday she was given an allowance from her parents. She was given $80 the first month and an increase of 5% every month. (a) Calculate the common ratio of this sequence. (b) Calculate the value of Vera s allowance after one year. 4. Javier bought a car with an original price of that depreciates by each year. (a) Calculate the common ratio of this sequence. (b) Calculate the value of Javier s car after six year.

5 Lesson 6-8: Homework Name Date 1. Prachi is on vacation in the United States. She is visiting the Grand Canyon. When she reaches the top, she drops a coin down a cliff. The coin falls down a distance of 5 metres during the first second, 15 metres during the next second, 25 metres during the third second and continues in this way. The distances that the coin falls during each second forms an arithmetic sequence. (a) Write down the common difference, d, of this arithmetic sequence. (b) Write down the distance the coin falls during the fourth second. 2. The population of big cats in Africa is increasing at a rate of 5 % per year. At the beginning of 2004 the population was (a) Write down the population of big cats at the beginning of (b) Find the population of big cats at the beginning of 2010.

6 3. A National Lottery is offering prizes in a new competition. The winner may choose one of the following. Option one: Option two: $250 in the first week, $450 in the second week, $650 in the third week, increasing by $200 each week for a total of 10 weeks. $10 in the first week, $20 in the second week, $40 in the third week continuing to double for a total of 10 weeks. (a) Based on the information, write an equation to represent the amount of money you will receive after n weeks for: (i) option one; (ii) option two. (b) Calculate the amount you receive in the tenth week, if you select (i) option one; (ii) option two. 4. The first three terms of a geometric sequence are. (a) Find the value of, the common ratio of the sequences (b) Find the exact value, in fraction form, of the 10 th term of the sequence. (c) Find the value of n for which.

7 5. The first term of an arithmetic sequence is 7 and the sixth term is 22. (a) Find the common difference. (b) Find the twelfth term. 6. The third term, u 3, of a geometric sequence is 36. The fifth term, u 5, of the sequence is 324. (a) Write down the common ratio of the sequence. (b) Find u 1.