Sequences and Series
Arithmetic
𝑎1 = first term
𝑎𝑛 = 𝑎𝑛−1 + 𝑑
𝑎𝑛 = 𝑎1 + (𝑛 − 1) ⋅ 𝑑
𝑛
𝑆𝑛 = (𝑎1 + 𝑎𝑛 )
2
… , 𝑎, 𝑏, 𝑐, …
𝑎+𝑐
𝑏=
2
Where do I belong?
𝑛)
𝐿𝑎𝑠𝑡 𝑡𝑒𝑟𝑚
#𝑎,(1
𝑛
𝑎
1
…
,
𝑏,
𝑐,
…
𝑎
−
𝑟
…
,
𝑎,
𝑏,
𝑐,
…
𝑛−1
𝑎
=
first
term
1 (𝑛
=
first
term
11
𝑎𝑛𝑆𝑆𝑛𝑎
𝑎
+
−𝑎1)
𝑎=
=
𝑎
⋅
𝑟
𝑆
=
=
(𝑎
+
)⋅ 𝑑
1
𝑛
1
1
𝑛
=
𝑎
+
𝑐
𝑛
𝐹𝑜𝑟𝑚𝑢𝑙𝑎
𝑏=
=
𝑎𝑐
2=𝑎𝑎±
1𝑛−1
−
𝑟+
𝑎𝑎𝑛𝑛𝐸𝑥𝑝𝑙𝑖𝑐𝑖𝑡
⋅ 𝑟𝑟𝑑
1
𝑏=
𝑛−1−
𝐹𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚 #
2
𝐴𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐 = 𝐿𝑖𝑛𝑒𝑎𝑟
𝐺𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐 = 𝐸𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙
𝑇𝑜𝑡𝑎𝑙 𝑇𝑒𝑟𝑚𝑠 = 𝑙𝑎𝑠𝑡 − 𝑓𝑖𝑟𝑠𝑡 + 1
Pg. 607 #1-32
Geometric
𝑎1 = first term
𝑎𝑛 = 𝑎𝑛−1 ⋅ 𝑟
𝑎𝑛 = 𝑎1 ⋅ 𝑟 𝑛−1
𝑎1 (1 − 𝑟 𝑛 )
𝑆𝑛 =
1−𝑟
𝑎1
𝑆=
1−𝑟
… , 𝑎, 𝑏, 𝑐, …
𝑏 = ± 𝑎𝑐
Sequence and Series
Describe
Arithmetic
Geometric
Summation
Modeling
10
10
10
10
10
20
20
20
20
20
30
30
30
30
30
40
40
40
40
40
50
50
50
50
50
Describe each using:
Sequence/Series; Arithmetic/Geometric; Finite/Infinite
2, 8, 14, 20, 26
Answer
Describe each using:
Sequence/Series; Arithmetic/Geometric; Finite/Infinite
2, 8, 14, 20, 26
Finite Arithmetic Sequence
Describe each using:
Sequence/Series; Arithmetic/Geometric; Finite/Infinite
2, 8, 32, 128, …
Answer
Describe each using:
Sequence/Series; Arithmetic/Geometric; Finite/Infinite
2, 8, 32, 128, …
Infinite Geometric Sequence
Describe each using:
Sequence/Series; Arithmetic/Geometric; Finite/Infinite
1 + 5 + 9 + ∙∙∙
Answer
Describe each using:
Sequence/Series; Arithmetic/Geometric; Finite/Infinite
1 + 5 + 9 + ∙∙∙
Infinite Arithmetic Series
Describe each using:
Sequence/Series; Arithmetic/Geometric; Finite/Infinite
16 + 8 + 4 + 2 + 1
Answer
Describe each using:
Sequence/Series; Arithmetic/Geometric; Finite/Infinite
16 + 8 + 4 + 2 + 1
Finite Geometric Series
Describe each using:
Sequence/Series; Arithmetic/Geometric; Finite/Infinite
∞
3(0.25)𝑛
𝑛=1
Answer
Describe each using:
Sequence/Series; Arithmetic/Geometric; Finite/Infinite
∞
3(0.25)𝑛
𝑛=1
Infinite Geometric Series
Write in summation notation
−31, −28, −25, −22, . . . , 2
Answer
Write in summation notation
−31, −28, −25, −22, . . . , 2
𝑎𝑛 = 𝑎1 + 𝑛 − 1 ⋅ 𝑑
2 = −31 + 𝑛 − 1 ⋅ 3
2 = −31 + 3𝑛 − 3
2 = −34 + 3𝑛
36 = 3𝑛
12 = 𝑛
12
−31 + (𝑛 − 1) ⋅ 3
𝑛=1
Or
12
3𝑛 − 34
𝑛=1
Write a recursive definition, explicit formula and find the
15 term for the sequence:
12, 21, 30, 39, . . .
Answer
Write a recursive definition, explicit formula and find the
15 term for the sequence:
12, 21, 30, 39, . . .
Recursive: 𝑎1 = 12; 𝑎𝑛 = 𝑎𝑛−1 + 9
Explicit: 𝑎𝑛 = 12 + (𝑛 − 1) ⋅ 9
𝑎15 = 12 + 15 − 1 ⋅ 9 = 138
Write a recursive definition, explicit formula and find the
18 term for the sequence:
45, 37, 29, 21, . . .
Answer
Write a recursive definition, explicit formula and find the
18 term for the sequence:
45, 37, 29, 21, . . .
Recursive: 𝑎1 = 45; 𝑎𝑛 = 𝑎𝑛−1 − 8
Explicit: 𝑎𝑛 = 45 + (𝑛 − 1) ⋅ (−8)
𝑎18 = 45 + 18 − 1 ⋅ −8 = −91
Find the missing term of the arithmetic sequence
… , −1,
, 11, . . .
Answer
Find the missing term of the arithmetic sequence
… , −1,
, 11, . . .
−1 + 11
=
=5
2
Find the missing terms of the arithmetic sequence
… , −13,
,
,
Answer
, 3, . . .
Find the missing terms of the arithmetic sequence
… , −13, −9, −5,
,
,−1, , 3, . . .
−13 + 3
=
= −5
2
−13 + (−5)
=
= −9
2
= −5 + 4 = −1
Write in summation notation
5, 10, 20, 40, … , 640
Answer
Write in summation notation
5, 10, 20, 40, … , 640
𝑎𝑛 = 𝑎1 ⋅ 𝑟 𝑛−1
640 = 5 ⋅ 2𝑛−1
128 = 2𝑛−1
𝑛 − 1 = log 2 128
𝑛−1=7
𝑛=8
8
5 ⋅ 2𝑛−1
𝑛=1
Write a recursive definition, explicit formula and find the
10 term for the sequence:
−3, 6, −12, 24, . . .
Answer
Write a recursive definition, explicit formula and find the
10 term for the sequence:
−3, 6, −12, 24, . . .
Recursive: 𝑎1 = −3; 𝑎𝑛 = 𝑎𝑛−1 ⋅ −2
Explicit: 𝑎𝑛 = (−3) ⋅ (−2)𝑛−1
𝑎10 = (−3) ⋅ (−2)10−1 = 1536
Write a recursive definition, explicit formula and find the
9 term for the sequence:
−2, −10, −50, −250, . . .
Answer
Write a recursive definition, explicit formula and find the
9 term for the sequence:
−2, −10, −50, −250, . . .
Recursive: 𝑎1 = −2; 𝑎𝑛 = 𝑎𝑛−1 ⋅ 5
Explicit: 𝑎𝑛 = (−2) ⋅ (5)𝑛−1
𝑎9 = −2 ⋅ 5
9−1
= −781250
Find the missing term of the geometric sequence
… , 3,
, 12, . . .
Answer
Find the missing term of the geometric sequence
… , 3,
, 12, . . .
= ± 12 ⋅ 3 = ±6
Find the missing terms of the geometric sequence
… , −20,
,
,
Answer
, −1.25, . . .
Find the missing terms of the geometric sequence
… , −20, ±10,, −5, ±2.5,
,
, −1.25, . . .
= ± −20 ⋅ −1.25 = ±5
= ± −20 ⋅ −5 = ±10
1
= −5 ⋅ ± = ±2.5
2
Evaluate the sum of each series
5 + 11 + 17 + ⋯ + 35
Answer
Evaluate the sum of each series
5 + 11 + 17 + ⋯ + 35
𝑎𝑛 = 𝑎1 + (𝑛 − 1) ⋅ 𝑑
35 = 5 + 𝑛 − 1 ⋅ 6
35 = 5 + 6𝑛 − 6
35 = 6𝑛 − 1
36 = 6𝑛
6=𝑛
𝑛
𝑆𝑛 = (𝑎1 + 𝑎𝑛 )
2
6
𝑆6 = (5 + 35) = 120
2
Evaluate the sum of each series
4 + 16 + 64 + ⋯ + 4096
Answer
Evaluate the sum of each series
4 + 16 + 64 + ⋯ + 4096
𝑎𝑛 = 𝑎1 ⋅ 𝑟 𝑛−1
4096 = 4 ⋅ 4𝑛−1
1024 = 4𝑛−1
𝑛 − 1 = log 4 1024
𝑛−1=5
𝑛=6
𝑎1 (1 − 𝑟 𝑛 )
𝑆𝑛 =
1−𝑟
4(1 − 46 )
𝑆6 =
= 5460
1−4
Evaluate the sum of each series
15
7𝑛 − 5
𝑛=4
Answer
Evaluate the sum of each series
15
7𝑛 − 5
𝑛=4
𝑎4 = 7 4 − 5 = 23
𝑎15 = 7 15 − 5 = 100
𝑛
𝑆𝑛 = (𝑎1 + 𝑎𝑛 )
2
𝑆12
12
=
(23 + 100) = 738
2
Evaluate the sum of each series
7
4(5)𝑛−1
𝑛=1
Answer
Evaluate the sum of each series
7
4(5)𝑛−1
𝑛=1
𝑎1 =
4(5)1−1 =
4
𝑎1 (1 − 𝑟 𝑛 )
𝑆𝑛 =
1−𝑟
4(1 − 57 )
𝑆6 =
= 78124
1−5
Evaluate the sum of each series
3
3𝑛2
𝑛=1
Answer
Evaluate the sum of each series
3
3𝑛2
𝑛=1
𝑎1 = 3(1)2 = 3
𝑎2 = 3(2)2 = 12
+ 𝑎3 = 3(3)2 = 27
42
The number of toy rockets made by an assembly line
for 8 hours forms an arithmetic sequence. If the line
produced 40 rockets in hour one and 43 rockets in hour
two, how many rockets will be produced in hour
seven?
How many rockets will be produced in one 8 hour
day?
Answer
The number of toy rockets made by an assembly line for
8 hours forms an arithmetic sequence. If the line
produced 40 rockets in hour one and 43 rockets in hour
two, how many rockets will be produced in hour seven?
How many rockets will be produced in one 8 hour day?
𝑎𝑛 = 𝑎1 + (𝑛 − 1) ⋅ 𝑑
𝑎𝑛 = 40 + 𝑛 − 1 ⋅ 3
𝑎7 = 40 + 7 − 1 ⋅ 3
= 58
𝑛
𝑆𝑛 = (𝑎1 + 𝑎𝑛 )
2
8
𝑆8 = (40 + 61) = 404
2
You invested money in a fund and each month
you receive a payment for your investment.
Over the first four months, you received $50,
$57, $64, $71. If this pattern continues, how
much will you receive in the 12th month and
who much will you receive for the entire year?
Write an explicit equation to model the
problem.
Answer
You invested money in a fund and each month you
receive a payment for your investment. Over the first
four months, you received $50, $57, $64, $71. If this
pattern continues, how much will you receive in the 12th
month and who much will you receive for the entire
year? Write an explicit equation to model the problem.
𝑎𝑛 = 𝑎1 + (𝑛 − 1) ⋅ 𝑑
𝑎𝑛 = 50 + 𝑛 − 1 ⋅ 7
𝑎12 = 50 + 12 − 1 ⋅ 7
= 127
𝑛
𝑆𝑛 = (𝑎1 + 𝑎𝑛 )
2
12
𝑆8 =
(50 + 127)
2
= 1062
You are trying to save $1500. You begin with $5 and
save $3 more than the previous week for 30 weeks.
Will you meet your goal?
• Write an explicit formula to model this problem
• What is the amount you will save in week 30?
• What is the total amount you will save over 30
weeks?
Answer
You are trying to save $1500. You begin with $5 and save
$3 more than the previous week for 30 weeks. Will you
meet your goal?
No
• Write an explicit formula to model this problem
• What is the amount you will save in week 30?
• What is the total amount you will save over 30 weeks?
𝑎𝑛 = 𝑎1 + (𝑛 − 1) ⋅ 𝑑
𝑛
𝑆𝑛 = (𝑎1 + 𝑎𝑛 )
2
𝑎𝑛 = 5 + 𝑛 − 1 ⋅ 3
𝑎30 = 5 + 30 − 1 ⋅ 3
= 92
𝑆30
30
=
(5 + 92)
2
= 1455
You saved $500 this year. Each year you plan to save
5% more than the previous year.
Write an explicit formula to model this situation.
How much will you save in the 8 year?
How much will you have saved totally over years,
assume you do not spend anything?
Answer
You saved $500 this year. Each year you plan to save 5% more than
the previous year.
Write an explicit formula to model this situation.
How much will you save in the 8 year?
How much will you have saved totally over years, assume you do
not spend anything?
𝑎1 (1 − 𝑟 𝑛 )
𝑆𝑛 =
𝑛−1
𝑎𝑛 = 𝑎1 ⋅ 𝑟
𝑎𝑛 = 500 ⋅ (1.05)𝑛−1
𝑎8 = 500 ⋅ (1.05)8−1
= 703.55
1−𝑟
500(1 − 1.058 )
𝑆8 =
1 − 1.05
= 4774.55
You drop a ball from a staircase that is 36
feet high. By the time you get down the
stairs to measure the height of the bounce,
the ball has bounced four times and has a
height of 2.25 feet after its fourth bounce.
How high did the ball bounce after it first
hit the floor? (hint the bouncing ball creates
a geometric sequence)
Answer
You drop a ball from a staircase that is 36 feet high.
By the time you get down the stairs to measure the
height of the bounce, the ball has bounced four
times and has a height of 2.25 feet after its fourth
bounce. How high did the ball bounce after it first
hit the floor?
18
36, 18, , 9, ,
, 2.25
= ± 36 ⋅ 2.25 = 9
= ± 36 ⋅ 9 = 18