1. Arithmetic Sequences Objective:
To find the next term in a sequence by looking for a pattern
To find the nth term of an arithmetic sequence
To find the position of a given term in an arithmetic sequence
To find arithmetic means
Arithmetic Sequences

2. Precalculus Unit 2 Sequences & Series
Do Now: Look at the pattern and write the next number or expression:
a) 1000, 500, 250, 125… 
b) 1, 2, 4, 7, 11, 16…
c) 1, -3, 9, -27…
d) 8, 3, -2, -7…
e) 2, 2 , 4, …
f) 7a + 4b, 6a + 5b, 5a + 6b, 4a + 7b…
g) 1, 3, 7, 11, 16…
Then determine: “just a sequence” OR Arithmetic OR Geometric
We will:
To find the next term in a sequence by looking for a pattern
To find the nth term of an arithmetic sequence
To find the position of a given term in an arithmetic sequence
To find arithmetic means

3. Sequence- a set of numbers {1, 3, 5, 7, …}
Vocabulary
Arithmetic Sequence- each term after the first is found by adding a constant, called the common difference, d, to the previous term
Geometric Sequence – each term after the first is found by MULTIPLYING a constant, called the common ratio, r, to get the next term
Sequence- a set of numbers {1, 3, 5, 7, …}
Terms- each number in a sequence
Common Difference- the number added to find the next term of an arithmetic sequence.
Common Ratio - number multiplied to find the next term of an geometric sequence

4. Arithmetic v. Geometric
DO NOW answers
Arithmetic v. Geometric
2, 5, 8, 11, 14….
You are ADDING to get the next term
Defining nth term:
In this case:
In words:
Each term equals the first term + the difference n-1 times.
3, 6, 12, 24, 48…
You are MULTIPLYING to get the next term
Defining nth term:
R is the common ratio.
In this case:
In words:
Each term equals the first term x the ratio n-1 times.

5. an-1 Formula for the nth term of an Arithmetic Sequence
nth term 1st term # of common
the term difference
trying to
RECURSIVE: A recursion formula for a sequence specifies tn as a function of the preceding term,
an-1
EXPLICIT : An explicit formula for a sequence specifies an as a function of n

6. Formula for the nth term of an Arithmetic Sequence nth term 1st term # of common Find the indicated term. the term difference 3) trying to find

7. Find the next four terms of each arithmetic sequence

8. Arithmetic & Geometric SequencEs
Arithmetic sequence: each term is formed by adding a constant to the previous term
Common difference: the amount being added or subtracted from one term in a sequence to the next
Geometric sequence: sequence in which each term is formed by multiplying the previous term by a constant
Common ratio: the amount being multiplied
T1 or a1: if you have a sequence or series, this is a symbol for the first term
Tn or an: this is a general term that represents the nth term.
Tn-1: this term represents the term that comes BEFORE

9. n tn
1 1
2 4
3 7
4 10
5 13
RECURSIVE Equation FOR THE ABOVE ARITHMETIC SEQUENCE WOULD BE:
tn = 3(n-1) + 1

10. What is the is the 22nd term? T22 = 3(22-1) + 1
what would the 11th term be?
  T11 = 3(n-1) + 1
 What is the is the 22nd term?
T22 = 3(22-1) + 1

11. Explicit formula:
Explicit formula: an equation that explains how to calculate a term in a sequence directly from its first term
Explicit equation for the above Arithmetic Sequence : tn = = 3(n) – 2
Calculate the 50th term of this sequence
Calculate the 150th term of this sequence

12. Which number term for this sequence will be 46?

13. Which number term for this sequence will be 46? 16

14. ANSWERS: Students find equations for the other two problems, and calculate the 50th and 100th term of each sequence…
5) : tn = (n) OR 3(n) -1
6) : tn = -2 – 3(n) OR (n) -2

15. Precalculus Unit 2 Sequences & Series
Do Now: take out hw and answer the following: identify the common sum/difference/ratio/divisor, and find the next 3 terms of each problem
a. 1, 4, 7, 10, 13….
b. 2, 5, 8, 11, 14….
c. -5, -8, -11, -14…
d. 1/2, 1, 2, 4, 8,...
e. 1/9, 1/3, 1, 3….
f. -2, 4, -8, 16…
CW: DO NOW part 2-VOCAB with ERITREA PARTNER!!
PUZZLE TIME..
We will:
find the next term in a sequence by looking for a pattern
find the nth term of an arithmetic sequence
find the nth term of an geometric sequence!!
HW: HANDOUT (do now) AND STUDY FOR QUIZ TOMORROW!!

16. Recursive vs. Explicit GEOMETRIC formulaS
N Tn
1 1/2
2 1
3 2
4 4
5 8
tn = t1(r)n RECURSIVE
formula
 What would the Recursive equation be for this sequence?
If the 7th term of this equation is 32, what would the 8th term be?

17. tn = t1(r)n-1
Find the recursion equation for the following 2 sequences
1/9, 1/3, 1, 3….
-2, 4, -8, 16…

18. Explicit GeoMETRIC formulaS
N Tn
1 1/2
2 1
3 2
4 4
5 8
tn = t0(r)n EXPLICIT formula What is the explicit equation for 1/2, 1, 2, 4, 8,...

19. Explicit GeoMETRIC formulaS
N Tn
1 1/2
2 1
3 2
4 4
5 8
tn = t0(r)n EXPLICIT
formula 
 Equation for 1/2, 1, 2, 4, 8, tn = ½ (2)n Calculate the 50th term of this sequence
Calculate the 150th term of this sequence

20. Using explicit Geometric formula
tn = t0(r)n AND in this case: tn = ½ (2)n Which number term for this sequence will be 128? Which number term for this sequence will be 256?

21. Using explicit Geometric formula
tn = t0(r)n
Find equations and calculate the 57th and 38th term of each series…
1/9, 1/3, 1, 3….
-2, 4, -8, 16…

22. Extra Credit
There is a new English Language school in Central Square in Cambridge with 26 students. Of each month, 3 new students enroll, in how many weeks will the school have

23. DO NOW:
A sequence contains t1 = 1, t2 = 2 and t5 = 16, Find the EQUATION (recursive AND EXPLICIT) for
Geometric sequence
Then find t10 for using recursive or explicit formula
Find t20
IF DONE-GOLD STAR:
Find the Arithmetic formulasequence (d = contains a decimal)
reminder;: hw: study + note card

24. DO NOW:
A sequence contains t1 = 1, t2 = 2 and t5 = 16, Find the EQUATION (recursive AND EXPLICIT) for
Geometric sequence
Then find t10 for using recursive or explicit formula
Find t20
Extra Credit : Find the Arithmetic formulasequence (d = contains a decimal)
reminder;: hw: study + note card

25. Complete each statement. 4)170 is the ____th term of –4, 2, 8

26. Find the indicated term. 5)

27. Find the missing terms in each sequence

28. Write an equation for the nth term of the arithmetic sequence

29. 11.2 Arithmetic Series Objective: To find sums of arithmetic series
To find specific terms in an arithmetic series
To use sigma notation to express sums
11.2 Arithmetic Series

31. Tn or an: this is a general term that represents the nth term.
Arithmetic sequence: each term is formed by adding a constant to the previous term
Common difference: the amount being added or subtracted from one term in a sequence to the next
Geometric sequence: sequence in which each term is formed by multiplying the previous term by a constant
Common ratio: the amount being multiplied
T1 or a1: if you have a sequence or series, this is a symbol for the first term
Tn or an: this is a general term that represents the nth term.
Tn-1: this term represents the term that comes BEFORE Tn

32. Vocabulary
Arithmetic Sequence- each term after the first is found by adding a constant, called the common difference, d, to the previous term
Sequence- a set of numbers {1, 3, 5, 7, …}
Terms- each number in a squence
Common Difference- the number added to find the next term of an arithmetic sequence.
Arithmetic Series- the sum of an arithmetic sequence
Series- the sum of the terms of a sequence
{ … +97}

33. Sum of an arithmetic Series
Arithmetic Sequence Arithmetic Series
4, 7, 10, 13,
-10, -4, (-4)+2
Sum of an arithmetic Series
Sum of Series # of series st term last term

34. 1) Find the sum of the 1st 50 positive even integers.

35. 2) Find the sum of the 1st 40 terms of an arithmetic series in which a1 = 70 and d = -21.

36. 3) A free falling object falls 16 feet in the first second, 48 feet in the 2nd second, 80 feet is the 3rd second, and so on. How many feet would a free-falling object fall in 20 seconds if air resistance is ignored?

37. 4) Find the 1st three terms of an arithmetic series where:

39. 6) Find the sum of each arithmetic series. 5 + 7 + 9 + … + 27

40. 11.3 Geometric Sequences Objective:
To find the nth term of a geometric sequence
To find the position of a given term in a geometric sequence
To find geometric means
11.3 Geometric Sequences

41. Geometric Sequence: multiplying each term by a common ratio (r).
Example: 3, 12, 48,… r = 4
Example: 100, 50, 25… r = ½
Formula:
nth term 1st term common ratio # of terms

42. Find the next 3 terms of each geometric sequence. 1) 2)

43. Find the 1st four terms of each geometric sequence

44. Find the nth term of each sequences. 5) 6)

45. Find geometric means. 7) 4, ____, ____, ____, 324 8) ____, ____, 12, ____, ____, 96

46. 11.4 Geometric Series Objective: To find sums of geometric series
To find the specific terms in a geometric series
To use sigma notation to express sums
11.4 Geometric Series

47. Geometric Sum Formula for Series
Sum of the nth terms 1st term common ratio nth term
Geometric Sequence Geometric Series
1, 3, 9, 27,
5, -10, 20, (-10) + 20

48. Find the sum of each geometric series.
1) …, n = 10 2) 2401 – – …, n = 5

49. Find the sum of each geometric series.
3) )

50. Find for each geometric series described.5)

51. Find for each geometric series described.6)

52. last term # 1st term # 1st term common ratio # of term
Sigma Notation:
last term # 1st term # 1st term common ratio # of term
Example: … to 7 terms

53. Express each series in sigma notation and find the sum
7) … to 10 terms

54. Express each series in sigma notation.8)

55. 11.5 Infinite Geometric Series
Objective:
To find the sums of infinite geometric series
11.5 Infinite Geometric Series

56. Sum of an Infinite Geometric Series
Sum 1st term common ratio

57. Find the sum of each infinite geometric series, if it exists

58. 2). An old grandfather clock is broken
2) An old grandfather clock is broken. When the pendulum is swung it follows a swing pattern of 25 cm, 20 cm, 16 cm, and so on until it comes to rest. What is the total distance the pendulum swings before coming to rest?

61. Evaluate. 6)

62. Evaluate. 7)

63. 11.6 Recursion and Special Sequences
Objectives:
To recognize and use special sequences
To iterate functions
11.6 Recursion and Special Sequences

64. Vocabulary
Recursive Formula- 2 parts: 1st the value of the first term, 2nd shows how to find each term from the term before it
Example (Fibonacci Sequence): 1, 1, 2, 3, 5, 8, …

65. Sequence Sequence Type
9, 13, 17, … arithmetic
7, 21, 63, … geometric
1, 1, 2, 3, 5, 8, … Fibonacci

70. The Binomial Theorem Objective:
To expand powers of binomials by using Pascal’s triangle
To find specific terms of binomial expansions

76. Evaluate by hand and using a calculator. 4)

80. Sum of an Infinite Geometric Series
Items on the Test
Arithmetic Sequence
Common Difference
Arithmetic Series
Geometric Sequence
Common Ratio
Geometric Series
Sigma Notation
Sum of an Infinite Geometric Series
Iterates
Binomial Theorem