2. Developing Explicit Formulas
Mountain Bike Rental
Developing Explicit Formulas

3. Rental Cost: Mountain Bike Rental
While on vacation, you want to rent a mountain bike.
Rental Cost:
There is a $10 rental fee, plus a $5 charge for each hour used.

4. 1. What would be the cost of renting the bike for 12 hours?
2. If you have a $100 budget for the bike rental, how many hours can you use it?

6. Rental Cost: Mountain Bike Rental
There is a $10 rental fee, plus a $5 charge for each hour used.

7. Let’s
take a
closer
look.

8. Record your data in a table for 0, 1, 2, … , 12 rental hours.

9. Record your data in a table for 0, 1, 2, … , 12 rental hours.
Graph the data.

10. Record your data in a table for 0, 1, 2, … , 12 rental hours.
Graph the data.
Describe the process in words.

11. Record your data in a table for 0, 1, 2, … , 12 rental hours.
Graph the data.
Describe the process in words.

14. c0 = ? cn = ? Mountain Bike Rental
Define a RECURSIVE FORMULA for this situation, which will predict the cost of the rental, for a given a number of hours.
c0 = ?
cn = ?
Define your variables.
Does c0 have meaning by itself? What is its meaning in the context of this situation?

15. cn = ? An EXPLICIT FORMULA gives a direct relationship
between two quantities.
cn = ?

16. The EXPLICIT FORUMLA for this situation can be found by examining the recursive pattern.
cn = ?

17. cn = 10 + 5n An EXPLICIT FORMULA 10 = the initial value
gives a direct relationship
between two quantities.
cn =
10 + 5n
10 = the initial value
5 = the rate of change

18. 1. What would be the cost of renting the bike for 12 hours?
2. If you have a $100 budget for the bike rental, how many hours can you use it?

19. 1. What would be the cost of renting the bike for 12 hours?
cn = n
c12 = ?

20. 2. If you have a $100 budget for the bike rental, how many hours can you use it?
cn = n
cn = 100
n = ?

21. 2. If you have a $100 budget for the bike rental, how many hours can you use it?
cn = n
cn = 100
n = ?
100 = n

23. Let’s examine the graph.

24. What type of graph is modeled by this data?

25. linear data What is the y-intercept of this line?
What is the line’s slope (rate of change)?
Compare these values to your recursive formula and your explicit formula.

27. c0 = 10 c0 = the initial value linear data The y-intercept of the line
(0, 10)
linear data
The y-intercept of the line
corresponds to the initial value of the recursive pattern.
c0 = the initial value

28. 3 hr
$15
1 hr $5
linear data
For an arithmetic sequence, the rate of change is the SLOPE of the line.
SLOPE:
5 dollars
1 hour

29. y = a + bx u0 = a un = a + bn un = un–1 + b Linear Equation
a = initial value
b = slope (rate of change)
Recursive Formula
u0 = a
Explicit Formula
un = a + bn
un = un–1 + b

30. y = a + bx y = 10 + 5x y = mx + b y = 5x + 10 Linear Equation
Important Note: The different uses of the variable b
other textbooks:
Linear Equation
y = mx + b
y = 5x + 10

32. Applying this to Sequences
Given the arithmetic sequence:
52, 73, 94, 115, …
1. Find u0
2. Represent the sequence three ways:
3. Find u29
4. Find n if un = 1186

33. Applying this to Sequences
52, 73, 94, 115, …
1. Find u0 n un 1 2 3 4 5 6
2. Find u29
3. Find n if un = 1186

34. Applying this to Sequences
52, 73, 94, 115, …
1. Find u0 n un 1 52 2 73 3 94 4
115 5 6
2. Find u29
3. Find n if un = 1186

35. Applying this to Sequences
52, 73, 94, 115, …
1. Find u0 n un 1 52 2 73 3 94 4
115 5
136 6
157
2. Find u29
3. Find n if un = 1186

36. Applying this to Sequences
52, 73, 94, 115, …
1. Find u0 n un 31 1 52 2 73 3 94 4
115 5
136 6
157
2. Find u29
3. Find n if un = 1186

38. Practice
Given the arithmetic sequence:
34, 53, 72, 91, …
1. Find u0
2. Represent the sequence three ways:
3. Find u40
4. Find n if un = 661

39. Practice
Given the arithmetic sequence:
205, 173, 141, 109, …
1. Find u0
2. Represent the sequence three ways:
3. Find u25
4. Find n if un = –275