Test: Chapter 8 Number patterns and recursion

Transcription

1 Student name: Multiple-choice questions (12 marks) 1 In the sequence 6, 13, 20, 27, the value of the common difference, d, is: A 21 B 7 C 7 D 21 E 34 2 Th e thirteenth term, t 13, in the sequence 63, 56, 49, 42, is: A 21 B 7 C 13 D 35 E Using the recurrence relation t 1 = 8, t n+1 = t n + 9, the sixth term would be: A 15 B 17 C 23 D 53 E 62 4 A canoe costs $40 to hire for the first day and $30 for each extra day. If C n is the cost of hiring the canoe for n days, the recurrence relation is: A C 0 = 40, C n+1 = C n + 30 B C 1 = 40, C n+1 = C n + 30 C C 0 = 40, C n = C n D C 1 = 30, C n+1 = C n + 40 E C 0 = 30, C n = C n

2 5 The common ratio, r, of the sequence 54, 72, 96, 128, is: A 18 B C D E 18 6 The recurrence relation for the sequence 9, 27, 81, is: A t 1 = 9, t n+1 = 9t n B t 1 = 9, t n+1 = t n + 9 C t 1 = 9, t n+1 = 3t n D t 1 = 9, t n+1 = t n + 18 E t 1 = 9, t n+1 = t n + 3, 7 In the sequence of patterns below pattern 1 used six sticks. Pattern 1 Pattern 2 Pattern 3 The number of sticks required to make Pattern 100 would be: A 402 B 406 C 598 D 600 E A 7% increase is made by using a common ratio of: A 0.07 B 1.07 C 1.7 D 7 E 107 2

3 9 A colony of frogs increases by 20% each year. If there were originally 500 frogs in the colony, the recurrence relation for the number of frogs F n after n years is: A F 1 = 500, F n+1 = 1.20F n B F 1 = 500, F n+1 = 0.20F n C F 0 = 500, F n+1 = 0.20F n D F 0 = 500, F n+1 = 1.20F n E F 1 = 500, F n = 1.20F n+1 10 The following recurrence relation can be used to model a compound interest investment of $2000, paying interest at the rate of 9% per annum. V 0 = 2000, V n+1 = 1.09V n After how many years will the value of the investment first exceed $5000? A 10 B 11 C 12 D 15 E In an arithmetic sequence t 4 = 15 and t 7 = 33. The value for t 11 is: A 15 B 33 C 48 D 57 E Th e first three terms in a sequence using the recurrence relation t 1 = 3, t n+1 = 2t n 5, are: A 3, 1, 1, B 1, 3, 11, C 2, 5, 3, D 3, 1, 3, E 3, 2, 5, 3

4 Short-answer questions (18 marks) n 1 Write the first 3 terms of the sequence t ( 2) n, n {1,2, 3,...}. (3 marks) n 2 Explain why t n:{2, 5, 8, } is an arithmetic sequence. (1 mark) 3 Find the rule (used to calculate the nth term) for the arithmetic sequence t n: {3, 7, 11, }. (3 marks) 4 Find the 10th term of the arithmetic sequence where the first term is 6 and the common difference is 4. (3 marks) 5 Explain why t n:{80, 40, 20, } is a geometric sequence. (2 marks) 4

5 6 Find the rule (used to calculate the nth term) for the geometric sequence t n :{3, 6, 12, }. (2 marks) 7 The three consecutive terms of the geometric sequence are 3, y, 27. Find the positive value of y. (1 mark) 8 Luke applied some hair shampoo to his hair. In each successive rinse, the amount of shampoo removed from Luke s hair followed the sequence: t n:{20 g, 10 g, 5 g, 2.5 g, }. Determine the total amount of shampoo removed from his hair after 5washes. (3 marks) 5

6 Extended-response questions (24 marks) 1 Steven accepted a job with a salary of $ in the first year and an increase of $7000 each following year. a Show his salary for each of the first 5 years in a table. b Use the table of values to plot the graph Comment on the form of the graph 6

7 Kathy accepted a job which paid $ in the first year and increased by 11% in each following year. c Show her salary for each of the first 5 years in a table. d Use the table of values to plot the graph e Comment on the form of the graph and give an explanation to support your answer 7

8 f After how many years does Kathy s salary exceed Steven s salary? Use mathematics to justify your answer. ( = 16 marks) 2 An oil well started producing 2000 barrels of oil per day. The rate at which oil production reduces each day is called the decline rate and is used to predict the productive life of the oil well. a Output for the first three days was recorded as: 2000, 1980, The decline rate is the percentage that production is reducing by each day. Calculate the decline rate for this oil well (correct to 2 decimal places). b State the recurrence relation for the oil production P n on the n th day of production. c What is the expected production for the well on the tenth day of operation (correct to 2 decimal places)? 8

9 d How many days of production will be possible before the daily output falls below 1700 barrels (to the nearest whole day)? ( = 8 marks) 9