The document discusses sequences and explicit formulas for sequences. It provides examples of using explicit formulas to find specific terms in sequences. Key concepts covered include terms, subscripts, explicit formulas, and notation for sequences. Examples show how to calculate terms by plugging values into explicit formulas and writing sequence terms with subscript notation.

2. Warm Up

3. Warm Up
1. Find the next three terms of the sequence:
81, 27, 9, 3,__, __, __

4. Warm Up
1. Find the next three terms of the sequence:
81, 27, 9, 3,__, __, __
2. Evaluate 4n - 6, when n =14.

5. Warm Up
1. Find the next three terms of the sequence:
81, 27, 9, 3,__, __, __
Divide the previous term by 3 to get 1, 1/3, 1/9
2. Evaluate 4n - 6, when n =14.

6. Warm Up
1. Find the next three terms of the sequence:
81, 27, 9, 3,__, __, __
Divide the previous term by 3 to get 1, 1/3, 1/9
2. Evaluate 4n - 6, when n =14.
4(14) - 6

7. Warm Up
1. Find the next three terms of the sequence:
81, 27, 9, 3,__, __, __
Divide the previous term by 3 to get 1, 1/3, 1/9
2. Evaluate 4n - 6, when n =14.
4(14) - 6
56 - 6

8. Warm Up
1. Find the next three terms of the sequence:
81, 27, 9, 3,__, __, __
Divide the previous term by 3 to get 1, 1/3, 1/9
2. Evaluate 4n - 6, when n =14.
4(14) - 6
56 - 6
50

10. 1.7 Explicit Formulas
for Sequences

11. 1.7 Explicit Formulas
for Sequences
E. Q. - How do we evaluate sequences?

13. Vocabulary

14. Vocabulary
• TERM - each number in a sequence

15. Vocabulary
• TERM - each number in a sequence
• SEQUENCE - function whose domain is
the set of natural numbers from 1 to n

16. Vocabulary
• TERM - each number in a sequence
• SEQUENCE - function whose domain is
the set of natural numbers from 1 to n
• SUBSCRIPT - number or variable that is
written below and to the right of another,
also called the index

17. Vocabulary
• TERM - each number in a sequence
• SEQUENCE - function whose domain is
the set of natural numbers from 1 to n
• SUBSCRIPT - number or variable that is
written below and to the right of another,
also called the index
• EXPLICIT FORMULA for the nth term -

18. Vocabulary
• TERM - each number in a sequence
• SEQUENCE - function whose domain is
the set of natural numbers from 1 to n
• SUBSCRIPT - number or variable that is
written below and to the right of another,
also called the index
• EXPLICIT FORMULA for the nth term -
• use it to calculate the nth term

19. Vocabulary
• TERM - each number in a sequence
• SEQUENCE - function whose domain is
the set of natural numbers from 1 to n
• SUBSCRIPT - number or variable that is
written below and to the right of another,
also called the index
• EXPLICIT FORMULA for the nth term -
• use it to calculate the nth term
• can calculate any term in the sequence

20. Vocabulary
• TERM - each number in a sequence
• SEQUENCE - function whose domain is
the set of natural numbers from 1 to n
• SUBSCRIPT - number or variable that is
written below and to the right of another,
also called the index
• EXPLICIT FORMULA for the nth term - n(n + 1)
2
• use it to calculate the nth term
• can calculate any term in the sequence
€

22. Examples

23. Examples
1. Use n(n + 1) to find the fifteenth rectangular number.
€

24. Examples
1. Use n(n + 1) to find the fifteenth rectangular number.
15(15 + 1)
€

25. Examples
1. Use n(n + 1) to find the fifteenth rectangular number.
15(15 + 1)
€ 15(16)

26. Examples
1. Use n(n + 1) to find the fifteenth rectangular number.
15(15 + 1)
€ 15(16)
240

27. Examples
1. Use n(n + 1) to find the fifteenth rectangular number.
15(15 + 1)
€ 15(16)
240
SEQUENCE NOTATION:

28. Examples
1. Use n(n + 1) to find the fifteenth rectangular number.
15(15 + 1)
€ 15(16)
240
SEQUENCE NOTATION: t20 = 440

29. Examples
1. Use n(n + 1) to find the fifteenth rectangular number.
15(15 + 1)
€ 15(16)
240
SEQUENCE NOTATION: t20 = 440 “t sub 20”

30. Examples
1. Use n(n + 1) to find the fifteenth rectangular number.
15(15 + 1)
€ 15(16)
240
SEQUENCE NOTATION: t20 = 440 “t sub 20”
means the 20th term is 440

32. 2. Consider the formula t n = 15 + 2(n −1) for integers n ≥ 1.
€

33. 2. Consider the formula tn = 15 + 2(n −1) for integers n ≥ 1.
a. what are the first four terms generated by it?
€

34. 2. Consider the formula tn = 15 + 2(n −1) for integers n ≥ 1.
a. what are the first four terms generated by it?
15 + 2(1-1) €
->

35. 2. Consider the formula tn = 15 + 2(n −1) for integers n ≥ 1.
a. what are the first four terms generated by it?
15 + 2(1-1) €
-> 15 + 2(0) ->

36. 2. Consider the formula tn = 15 + 2(n −1) for integers n ≥ 1.
a. what are the first four terms generated by it?
15 + 2(1-1) €
-> 15 + 2(0) -> 15 + 0 = 15

37. 2. Consider the formula tn = 15 + 2(n −1) for integers n ≥ 1.
a. what are the first four terms generated by it?
15 + 2(1-1) €
-> 15 + 2(0) -> 15 + 0 = 15
VOLUNTEERS FOR THE NEXT THREE???

38. 2. Consider the formula tn = 15 + 2(n −1) for integers n ≥ 1.
a. what are the first four terms generated by it?
15 + 2(1-1) €
-> 15 + 2(0) -> 15 + 0 = 15
VOLUNTEERS FOR THE NEXT THREE???
2. b. Find t 8 .
€

39. 2. Consider the formula tn = 15 + 2(n −1) for integers n ≥ 1.
a. what are the first four terms generated by it?
15 + 2(1-1) €
-> 15 + 2(0) -> 15 + 0 = 15
VOLUNTEERS FOR THE NEXT THREE???
2. b. Find t 8 . 15 + 2 (8 -1 )
€

40. 2. Consider the formula tn = 15 + 2(n −1) for integers n ≥ 1.
a. what are the first four terms generated by it?
15 + 2(1-1) €
-> 15 + 2(0) -> 15 + 0 = 15
VOLUNTEERS FOR THE NEXT THREE???
2. b. Find t 8 . 15 + 2 (8 -1 )
15 + 2(7)
€

41. 2. Consider the formula tn = 15 + 2(n −1) for integers n ≥ 1.
a. what are the first four terms generated by it?
15 + 2(1-1) €
-> 15 + 2(0) -> 15 + 0 = 15
VOLUNTEERS FOR THE NEXT THREE???
2. b. Find t 8 . 15 + 2 (8 -1 )
15 + 2(7)
15 + 14
€

42. 2. Consider the formula tn = 15 + 2(n −1) for integers n ≥ 1.
a. what are the first four terms generated by it?
15 + 2(1-1) €
-> 15 + 2(0) -> 15 + 0 = 15
VOLUNTEERS FOR THE NEXT THREE???
2. b. Find t 8 . 15 + 2 (8 -1 )
15 + 2(7)
15 + 14
29
€

44. 3. Suppose you drop a ball from the top of a 50
foot wall. On each bounce the ball rises to 75%
of the previous height. The heights form a
sequence.

45. 3. Suppose you drop a ball from the top of a 50
foot wall. On each bounce the ball rises to 75%
of the previous height. The heights form a
sequence.
What are the first three bounce heights and after how many
bounces does the ball rise < 10 feet? YOU TRY!!!

46. 3. Suppose you drop a ball from the top of a 50
foot wall. On each bounce the ball rises to 75%
of the previous height. The heights form a
sequence.
What are the first three bounce heights and after how many
bounces does the ball rise < 10 feet? YOU TRY!!!
HINT: take .75 times the previous height to get
the new one

47. 3. Suppose you drop a ball from the top of a 50
foot wall. On each bounce the ball rises to 75%
of the previous height. The heights form a
sequence.
What are the first three bounce heights and after how many
bounces does the ball rise < 10 feet? YOU TRY!!!
HINT: take .75 times the previous height to get
the new one
37.5 ft, 28.125 ft, 21.09375 ft and 6 bounces!

49. LAST ONE!!!

50. LAST ONE!!!
Consider the sequence t, squares of consecutive positive
integers.

51. LAST ONE!!!
Consider the sequence t, squares of consecutive positive
integers.
a. Give an explicit formula for the sequence.

52. LAST ONE!!!
Consider the sequence t, squares of consecutive positive
integers.
a. Give an explicit formula for the sequence.
2
tn = n
€

53. LAST ONE!!!
Consider the sequence t, squares of consecutive positive
integers.
a. Give an explicit formula for the sequence.
2
tn = n
b. What is the value of t sub 4?
€

54. LAST ONE!!!
Consider the sequence t, squares of consecutive positive
integers.
a. Give an explicit formula for the sequence.
2
tn = n
b. What is the value of t sub 4?
€ 1, 4, 9, 16

55. LAST ONE!!!
Consider the sequence t, squares of consecutive positive
integers.
a. Give an explicit formula for the sequence.
2
tn = n
b. What is the value of t sub 4?
€ 1, 4, 9, 16
c. What is the value of t 250 ?
€

56. LAST ONE!!!
Consider the sequence t, squares of consecutive positive
integers.
a. Give an explicit formula for the sequence.
2
tn = n
b. What is the value of t sub 4?
€ 1, 4, 9, 16
c. What is the value of t 250 ?
2
250 = 62,500
€ €

57. Homework
Page 46 & 47
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