Chapter 1 Study Guide

Use these practice problems to help you study for your test. Answers are provided at the end.

Fig. 1 Fig. 2 Fig. 3

1.

  1. List the number of squares in figure 1 – 10
  1. Write a recursive formula that generates the sequence found in part a.
  1. How many squares are in figure 20?
  1. Which figure has 71 squares?
  1. List the number of segments in figure 1 – 10
  1. Write a recursive formula that generates the sequence found in part e.
  1. How many segments are in figure 20?
  1. Which figure has 250 segments?

2.

a. List the number of line segments in figures 1 – 5

  1. Write a recursive formula to represent the sequence.
  1. Find the number of line segments in figure 18.
  1. Which figure will have 44 segments?

3.

a. List the number of line segments in Figures 1-7.

b. Write a recursive formula that generates the sequence you found in part a.

c. How many line segments are in Figure 21?

d. Which figure has 89 line segments?

4. Consider the sequence 15, 11.25, 8.4375,

a. Is it arithmetic or geometric? Justify your answer.

b. Write a recursive formula for the sequence. Use u1 to represent the starting term.

c. What is the 6th term of the sequence?

d. Which term of the sequence is the first to be less than .1?

5. Consider the sequence 1000, 981, 962, 943

a. Is it arithmetic or geometric? Justify your answer.

b. Write a recursive formula for the sequence. Use u1 to represent the starting term.

c. What is the 6th term of the sequence?

d. Which term of the sequence is the first to be less than 850?

6. Consider the sequence4, 7, 10, 13…

a. State whether the above sequence is arithmetic or geometric. Justify your answer.

b. Write the recursive formula to represent the sequence. Use u1 as the starting term.

c. What is the 10th term in the sequence.

d. Which term is the first to be greater than 100?

7. Consider the sequence48, 12, 3, 0.75…

a. State whether the above sequence is arithmetic or geometric. Justify your answer.

b. Write the recursive formula to represent the sequence. Use u1 as the starting term.

c. What is the 8th term in the sequence.

d. Which term is the first to be less than 0.05?

8. Consider the sequence 5, 2.5, 1.25, 0.625, 0.3125,....

a. Is the sequence arithmetic or geometric? Justify your answer.

b. Write the recursive formula for this sequence. Use u1 to represent the first term.

c. What is the 9th term of the sequence?

  1. Which term of the sequence is the first to be less than 0.002?

9. An insect zoo has 7.8 million bugs. Each year 5% of the bugs die and they bring in 100,000 more bugs. How many bugs will there be after 3 years? (Honors: How many bugs will there be in the long run?)

10. A small planet far, far away named Roscolian had a population of 5.3 million aliens. Each year 4% of the population dies or leaves the planet for a fresh start somewhere else. Also, each year, 145,000 new cute baby aliens are born or other aliens will immigrate to Roscolian. What will the population be in 5 years? (Honors: Find the long-run value.)

11. A small country has a population of 2.3 million people. Each year about 5% of the pervious year's population dies or leaves the country and about 100,000 people are born or immigrate to the country. If this pattern continues, what will the population be in 5 years? (Honors: What will the population be in the long run?)

12. Write a recursive formula for a sequence whose graph fits the given description.
a. nonlinear decreasing toward zero. b. Nonlinear and increasing to zero

c. Linear and decreasingd. Nonlinear and decreasing to negative infinity.

e. Linear and increasingf. Constantg. nonlinear and increasing to infinity.

h. nonlinear and decreasing to anything but zero

i. nonlinear and increasing with a long-run value not infinity or zero

j. nonlinear and decreasing with a long-run value not zero or infinity

13. Describe what the graph for each recursive formula should look like (linear/nonlinear, increasing/decreasing). Then identify the sequence as arithmetic or geometric.

a. a0 = 25 and an=an-1 + 17.6 where n≥1

b. a0 = 25 and an = 0.4*an-1 where n≥1

c. a0 = 25 and an = an-1 - 14 where n≥1

d. a0 = 25 and an = 4*an-1 where n≥1

14. George’s principal in the bank is $3,000. The interest rate is 7% per year.

a. How much total money will he have after year 1?

  1. Write a recursive formula for the total amount of money he will have after each year.
  1. How many years will it take for George to have $4,000?

15. Lisa is on a desert island with 120 coconuts. She eats 9% of her coconuts each day.

a. Write a recursive formula for the total amount of coconuts she will have left.

  1. When will she have less than 10 coconuts?

16. A 2010 BMW 7 Series 4-door 750LixDrive Sedan costs $88,900.00 brand new. The value of the car depreciates by 15% each year

a. Write a recursive formula

b. What is the value of the car after 5 years?

17. You deposit $3,000 into a bank account that offers a monthly interest rate of 3.5%.

a. Write a recursive formula.

b. When will her balance be over $4000 dollars?

18. Find the % increase or decrease for each problem

a. From 40 to 50b. From 90 to 79c. From 20 to 90

d. From 80 students to 135 studentse. From 77 GPA to 84 GPA

f. From 45 brownies to 4 browniesg. From 24 miles to 57 miles

h. From 45 pints to 13 pints i. From 4048 minutes to 7548 minutes

j. From 326 ft to 241 ft k. From 45 ft to 92 ftl. From 309 grams to 299 grams

19. You are offered 2 jobs! Lucky you! Both jobs offer a starting salary of $30,000 a year. Company 1 offers a $650 raise each year. Company 2 offers a 2% raise each year.

a. Which offer should you take?

b. Write a recursive formula for both companies.

c. Make a table displaying the first 5 years at each company

d. Would Company 2’s salary ever exceed Company 1’s? If so, when?

e. What issues could effect your decision to choose a company?

20. Always, Sometimes, Never

a. If the common difference of an arithmetic sequence is 5, you ______subtract 5 from the previous to find the next term.

b. If the common ratio of a geometric sequence is a 4, you ______multiply by 4 from the previous term to find the next term.

c. (Honors) A shifted geometric sequence ______has a long-run value.

d. In a discrete graph, the points should ______be connected with a line.

e. A fractal is ______a geometric design that infinitely occurs.

f. The principal of a bank account is _____ 0.

g. Interest from a bank account ______makes the value increase.

h. The initial term of an sequence is _____ u1.

i. In a recursive formula, there is _____ an initial term and ____ a recursive rule

j. A decay or growth sequence is _____ arithmetic.

k. (Honors) The long-run (limit) value of a shifted geometric sequence ______hits the value.

21. Match each definition with a vocabulary word.

1. ______/ A sequence that has a common difference
2. ______/ The initial investment
3. ______/ A geometric sequence that also includes addition or subtraction(Honors)
4. ______/ A graph consisting of points
5. ______/ The quotient of the now term divide and the previous term
6. ______/ A geometric sequence that decreases
7. ______/ A shape that is split into parts, each of which is a reduced size and exact copy of the whole
8. ______/ Interest earned
Honors
9. ______/ Is a number that the sequence gets closer and closer to but never reaches
10. ______/ A formula that defines a sequence (It has a starting term and a rule)
11. ______/ A geometric sequence that increases
12. ______/ A sequence that has a common ratio
13. ______/ A number in the sequence
14. ______/ An ordered list of numbers
15. ______/ Points are on a straight line
16. ______/ Part of the recursive formula that uses the previous term (or terms) to find the next term
17. ______/ Process in which each step of a pattern is dependent on the step or steps before it
18. ______/ The previous term subtracted from the now term (now – previous)
19. ______/ The nth term of the sequence
  1. Accrued Interest
  2. Arithmetic Sequence
  3. Decay
  4. Difference
  5. Discrete Graph
  6. Fractal
  7. General Term
/
  1. Geometric Sequence
  2. Growth
  3. Limit
  4. Linear
  5. Principal
  6. Ratio
/
  1. Recursion
  2. Recursive Formula
  3. Recursive Rule
  4. Shifted Geometric Sequence
  5. Sequence
  6. Term

22. Honors: Write a recursive formula for the following sequence. Use u1 as the initial term.

14, 9.64, 4.1464, -2.775536, -11.497175

Answers

1. a. 1, 3, 5, 7, 9, 11, 13, 15, 17, 19...b. u1= 1; un=un-1+2 where n2c. 39

d. 35e. 4, 10, 16, 22, 28, 34, 40, 46, 52, 58...f. u1=4; un=un-1+6 where n2

g. 118h. 42

2. a. 4, 6, 8, 10, 12...b. u1 = 4;un=un-1+2 where n2c. 38d. 21

3. a. 3, 5, 7, 9, 11, 13, 15b. u1=3; un=un-1+2 where n2c. 43d. 44

4. a. Geometric; the common ratio is .75b. u1=15; un=un-1(.75) where n2

c. 3.56d. 19

5. a. Arithmetic; the common difference is -19b. u1=1000; un=un-1-19 where n2

c. 905d. 9

6. a. Arithmetic; the common difference is 3b. u1=4; un=un-1+3 where n2

c. 31d. 34

7. a. Geometric; the common ratio is 0.25b. u1=48; un=un-1(.25) where n2

c. 0.0029d. 6

8. a. Geometric; the common ratio is 0.5b. u1=5; un=un-1(0.5) where n2

c. 0.01953d. 13

9. 6,972,775 bugs; Honors: 2,000,000 bugs

10. 4,990,749 aliens; Honors: 3,625,000 aliens

11. 2,130,000 people; 2,000,000 people

12.(Answers may vary)

13. a. Arithmetic; Increasing; Linearb. Geometric; Decreasing; Nonlinear

c. Arithmetic; Decreasing; Lineard. Geometric; Increasing; Nonlinear

14. a. $3,210b. u0=3000; un=un-1(1.07) where n1c. 5 years

15. a. u0=120; un=un-1(.91) where n1b. 27

16. $39,445.40

17. a. u0=3000; un=un-1(1.035) where n1

18. a. 25% increaseb. 13% decreasec. 350% increased. 68.75% increase

e. 9% increasef. 92% decreaseg. 138% increaseh. 71% decrease

i. 86% increasej. 26% decreasek. 104% increasel. 3% decrease

19. a. Company 2b. Company 1: u0=30,000;un=un-1 + 650 where n1

Company 2: u0=30,000; un=un-1(1.02) where n1

c.

Year / 0 / 1 / 2 / 3 / 4 / 5
Company 1 / 30000 / 30650 / 31300 / 31950 / 32600 / 33250
33900 / 30000 / 30600 / 31212 / 31836.24 / 32472.96 / 33122.42
  1. Yes, after year 9. You would make $35,850 at Company 1 and $35,852.78 at Company 2
  2. Answers may vary

20. a. Neverb. Alwaysc. (Honors) Sometimesd. Nevere. Always

f. Sometimesg. Alwaysh. Sometimesi. Always; Alwaysj. Never

k. (Honors) Never

21.1. B2. L3. Q4. E5. M6. C7. F8. A9. (Honors) J10. O11. I

12. H13. S14. R15. K16. P17. N18. D19. G

22. u1 = 14

un = un-1(1.26) + 8 where n 2