SlideShare a Scribd company logo
1 of 80
Download to read offline
Arithmetic Sequence
and Arithmetic Series
1
2
At the end of the lesson, the learners are expected to:
1) Generate and describe patterns using symbols and mathematical
expressions;
2) Find the next few terms and the nth term of the given sequence;
3) Define and describe an arithmetic sequence;
4) Enumerate the next few terms and the nth term of an arithmetic
sequence;
5) Insert means between two given terms of an arithmetic sequence;
6) Find the sum of the first n terms of an arithmetic sequence; and
7) Solve problems involving arithmetic sequence.
3
4
5
6
1. O T T F F S S E N _?
T E F S
2. R O Y B G I _?
T U V W
3. J F M A M J J A S O N _?
A B C D
4. T Q P H H O N _?
D U H I
Direction: Click the letter of your answer. 7
11
12
1) 80, 40, 20, 10, 5, __?
2) 1, 8, 27, 64, 125, 216, __?
3) 1, 4, 9, 16, 25, 36, 49, __?
343 512 729 810
56 64 72 81
0
1
2
1
5
2
13
4) 1, 2, 3, 5, 8, 13, 21, __?
5) -2, -5, -8, -11, -14, __?
6) 7, 11, 15, 19, 23, 27, __?
-17 -19 -21 -23
29 30 31 32
31 32 33 34
14
19
20
21
22
23
24
25
26
27
28
29
30
31
An ordered list of numbers, such as
10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, …
is called a sequence.
A number sequence is a list of numbers having a
first number, a second number, a third number, and
so on, called the terms of the sequence.
32
FINITE AND INFINITE
SEQUENCES
• 10, 15, 20, 25, 30
• There are only a
finite number of
terms.
• They have a last
term.
• 11, 22, 33, 44, 55,
…
• There are infinite
number of terms.
• They do not have a
last term.
33
Sequences are usually given by stating their
general or nth term.
NUMBER SEQUENCE
Example:
Consider the sequence given 𝒂 𝒏 = 𝟑𝒏 + 𝟐
The first five terms of the sequence are
𝒂 𝟏 = 𝟑 𝟏 + 𝟐 = 𝟑 + 𝟐 = 𝟓,
𝒂 𝟐 = 𝟑 𝟐 + 𝟐 = 𝟔 + 𝟐 = 𝟖,
𝒂 𝟑 = 𝟑 𝟑 + 𝟐 = 𝟗 + 𝟐 = 𝟏𝟏,
𝒂 𝟒 = 𝟑 𝟒 + 𝟐 = 𝟏𝟐 + 𝟐 = 𝟏𝟒, 𝒂𝒏𝒅
𝒂 𝟓 = 𝟑 𝟓 + 𝟐 = 𝟏𝟓 + 𝟐 = 𝟏𝟕
34
a. 𝒂 𝒏 = 𝟐𝒏 𝟐
for 𝟏 ≤ 𝒏 ≤ 𝟓
b. 𝒂 𝒏 = 𝟐𝒏 − 𝟏 for 𝟏 ≤ 𝒏 ≤ 𝟓
c. 𝒂 𝒏 = (−𝟏) 𝒏
(𝒏 − 𝟑) 𝟐
for 𝟏 ≤ 𝒏 ≤ 𝟒
List all the indicated terms of each finite
sequence.
NUMBER SEQUENCE
35
37
QUESTION:
What do you observe in the following pictures
38
39
Joey’s school offered him a scholarship grant of ₱
3,000.00 when he was in Grade 7 and increased
the amount by ₱ 500 each year till Grade 10. The
amounts of money (in ₱) Joey received in Grade 7,
8, 9, and 10 were respectively:
3000, 3500, 4000, and 4500
Each of the numbers in the list is called a term.
Note: We find that the succeeding terms are
obtained by adding a fixed number.
40
In a savings scheme, the amount triples after every
7 years. The maturity amount (in ₱) of an
investment of ₱ 6,000.00 after 7, 14, 21 and 28
years will be, respectively:
18000, 54000, 162000, 486000
Note: We find that the succeeding terms are
obtained by multiplying with a fixed number.
41
NUMBER PATTERNS
The number of unit squares in a square with sides
1, 2, 3, 4, ... units are respectively 1, 4, 9, 16, ....
Note: We can observe that 1=12, 4=22, 9=32, 16=42, ...
Thus, the succeeding terms are squares of consecutive
numbers.
42
43
ARITHMETIC SEQUENCE
Consider the following lists of numbers :
2, 4, 6, 8, 10, ....
15, 12, 9, 6, 3, ....
-5, -4,-3, -2, -1....
6, 6, 6, 6, 6, 6, ....
each term is obtained by adding 2 to
the previous term
each term is obtained by adding -3
to the previous term
each term is obtained by adding 1 to
the previous term
each term is obtained by adding 0 to
the previous term
44
45
1) 23, 38, 53, __ , 83, 98
2) 45, 37, __ , 21, 13, 5
3) -13, -6, __ , 8, 15, 22
27 28 29 30
0 1 -2 -3
63 68 73 78
46
4) __ , 23, 32, 41, 50, 59
5) -12, -7, -2, 3, 8, ___
6) 10, __ , 32, 43, 54, 65
10 11 12 13
21 23 25 27
10 12 14 16
47
The constant is called the common difference, denoted with
the letter d, referring to the fact that the difference between
two successive terms yields the constant value that was
added. To find the common difference, subtract the first term
from the second term.
In mathematics, an arithmetic sequence or arithmetic
progression is a sequence of numbers where each term
after the first term is obtained by adding the same
constant (always the same).
52
Examples:
5, 10, 15, 20, 25, …
3, 7, 11, 15, …
20, 18, 16, 14, …
-7, -17, -27, -37, …
d=5
d=4
d=-2
d=-10
53
2, 5, 8, 11, 14 …
In this sequence, notice that the common difference is
3.
In order to get to the next term in the sequence, we
will add 3; so, a recursive formula for this sequence
is:
31  nn aa
The first term in the sequence is a1 (sometimes just
written as a).
Example:
54
2, 5, 8, 11, 14 …
+3 +3 +3 +3
Every time you want another term in the said sequence of
numbers, you have to add d. This means that the second
term was the first term plus d. The third term is the first
term plus d plus d (added twice). The fourth term is the
first term plus d plus d plus d (added three times). So,
you can see to get the nth term we have to take the first
term and add (n - 1) times d.
𝑑 = 3
 dnaan 1
𝑎 = 2
  1715231626 a 55
To find any term of an arithmetic sequence:
where:
a1 is the first term of the sequence,
d is the common difference,
n is the number of the term to find.
ARITHMETIC SEQUENCE
56
Find the nth term of the arithmetic sequence
5, 8, 11, 14, …
We know that 𝒂 𝟏 = 𝟓 𝒅 = 𝟑
Substituting in the formula, we obtain
𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑
𝑎 𝑛 = 5 + 𝑛 − 1 3
𝑎 𝑛 = 5 + 3𝑛 − 3
𝒂 𝒏 = 𝟐 + 𝟑𝒏
57
Find the 18th term of the arithmetic sequence
21, 24, 27, 30, 33, …
Note that 𝒂 𝟏 = 𝟓, 𝒅 = 𝟑, 𝒏 = 𝟏𝟖. Using the formula
for the general term of an arithmetic sequence, we have
𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑
𝑎 𝑛 = 21 + 18 − 1 3
𝑎 𝑛 = 21 + (17)3
𝒂 𝒏 = 𝟕𝟐
Thus, 72 is the 18th term of the sequence.
58
In the arithmetic sequence 10, 16, 22, 28, 34, … , which
term is 124?
The problem asks for n when 𝑎 𝑛 = 124. From the given
sequence 𝒂 𝟏 = 𝟏𝟎, 𝒅 = 𝟔, 𝒂 𝒏 = 𝟏𝟐𝟒. Substitute these
values in the formula to get
𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑
124 = 10 + 𝑛 − 1 6
124 = 10 + 6𝑛 − 6
120 = 6𝑛
𝟐𝟎 = 𝒏 Thus, 124 is the 20th term.
59
Find the first term of the arithmetic sequence with a
common difference of 6 and whose 20th term is 965?
Since 𝒂 𝟐𝟎 = 𝟗𝟔𝟓, 𝒅 = 𝟔, 𝒏 = 20 , we have
𝑎20 = 𝑎1 + 𝑛 − 1 𝑑
965 = 𝑎1 + 20 − 1 6
965 = 𝑎1 + 19 6
965 = 𝑎1 + 114
𝑎1 = 965 − 114 = 851
Thus, 851 is the first term.
60
Find the 15th term of the arithmetic sequence whose first
term is 7 and whose 5th term is 19.
Find the common difference, d by substituting 𝑎1 =
7 and, 𝑎5 = 19 . Thus,
𝑎5 = 𝑎1 + 𝑛 − 1 𝑑
19 = 7 + 5 − 1 𝑑
19 = 7 + 4𝑑
12 = 4𝑑
3 = 𝑑
Now use 𝒂 𝟏 = 𝟕, 𝒅 = 𝟑, 𝒏 = 𝟏𝟓
𝒂 𝟏𝟓 = 𝟕 + 𝟏𝟓 − 𝟏 𝟑 = 𝟒𝟗
Thus, the 15th term is 49.
61
Insert two arithmetic means between 8 and 17.
Our task here is to find 𝑎2 and 𝑎3 such that 8, 𝑎2, 𝑎3,
17 form an arithmetic sequence.
𝑎1 = 8 𝑎4 = 17
𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑
17 = 8 + 4 − 1 𝑑
17 = 8 + 3𝑑
15 = 3𝑑
5 = 𝑑
We need 𝑎2 and 𝑎3 . Hence, 𝑎2 = 8 + 3 = 𝟏𝟏 and
𝑎3 = 8 + 3 + 3 = 𝟏𝟒
62
A conference hall has 33 rows of seats. The last row
contains 80 seats. If each row has two fewer seats than
the row behind it. How many seats are there in the first
row?
We know that 𝒂 𝟑𝟑 = 𝟖𝟎, 𝒅 = 𝟐, 𝒏 = 33 . Using the formula
for the general term of an arithmetic sequence, we have
𝑎33 = 𝑎1 + 𝑛 − 1 𝑑
80 = 𝑎1 + 33 − 1 2
80 = 𝑎1 + 32 2
80 = 𝑎1 + 64
𝑎1 = 80 − 64 = 16
Therefore, there are 16 seats in the first row.
63
How many numbers between 7 and 500 are divisible by
5?
The common difference, d, is 5. Since we want numbers
that are divisible by 5, then we let 𝒂 𝟏 = 𝟏𝟎, 𝒂 𝒏 = 𝟓𝟎𝟎.
Substitute these values into the general term.
𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑
500 = 10 + 𝑛 − 1 5
500 = 10 + 5𝑛 − 5
495 = 5𝑛
𝟗𝟗 = 𝒏 There are 99 numbers between 7 and 500 that are
divisible by 5.
64
65
1. 1, 2, 3, 4, … and 1+2+3+4+…
2. 2, 4, 6, 8, … and 2+4+6+8+…
3. 5, 10, 15, … and 5+10+15+…
4. 3, 7, 11, 15, … and 3+7+11+15+…
5. 4, 8, 12, 16, … and 4+8+12+16+…
Each indicated sum of the terms of an
arithmetic sequence is an arithmetic series.
66
Do you know this?
The story is told of a grade school teacher In the 1700’s
that wanted to keep her class busy while she graded
papers so she asked them to add up all of the numbers
from 1 to 100. These numbers are an arithmetic sequence
with common difference 1. Carl Friedrich Gauss was in
the class and had the answer in a minute or two
(remember no calculators in those days). This is what he
did:
1 + 2 + 3 + 4 + 5 + . . . + 96 + 97 + 98 + 99 + 100
sum is 101
sum is 101
67
The formula for the sum of n terms is:
 nn aa
n
S  1
2
n is the number of terms so
𝒏
𝟐
would be the number of pairs
Let’s find the sum of 1 + 3 +5 + . . . + 59
ARITHMETIC SERIES
first term last term
68
 nn aa
n
S  1
2
Let’s find the sum of 1 + 3 +5 + . . . + 59
 12 nThe common difference is 2 and
the first term is 1, so:
Set this equal to 59 to find n. Remember n is the term number.
𝟐𝒏 − 𝟏 = 𝟓𝟗 𝒏 = 𝟑𝟎 So there are 30 terms to sum up.
  900591
2
30
30 S
first term last term
69
To find the sum of a certain number of
terms of an arithmetic sequence:
where:
Sn is the sum of n terms (nth partial sum),
a1 is the first term,
an is the nth term.
70
To find the sum of a certain number of
terms of an arithmetic sequence:
where:
Sn is the sum of n terms (nth partial sum),
a is the first term,
n is the “rank” of the nth term
d is the common difference
71
Find the sum of the first ten positive integers.
a1 = 1 n = 10
Illustrative Example
a10 = 10
𝑆10 =
𝑛
2
(𝑎1 + 𝑎 𝑛)
𝑆10 =
10
2
(1 + 10)
𝑆10 = 5 (11)
𝑆10 = 𝟓𝟓
72
Find the sum of the first 15 terms of the arithmetic
sequence if the first term is 11 and the 15th term
is 109.
a1 = 11 n = 15
Illustrative Example
a15 = 109
𝑆15 =
𝑛
2
(𝑎1 + 𝑎 𝑛)
𝑆15 =
15
2
(11 + 109)
𝑆15 =
15
2
(120)
𝑆15 = 𝟗𝟎𝟎
73
Find the sum of all the odd integers from 1 to
99.
a1 = 1 d = 2
Illustrative Example
Here, a10 = 99
𝑆50 =
𝑛
2
(𝑎1 + 𝑎 𝑛)
𝑆50 =
50
2
(1 + 99)
𝑆50 = 25 (100)
𝑆50 = 𝟐, 𝟓𝟎𝟎
𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑
99 = 1 + 𝑛 − 1 2
99 = 1 + 2𝑛 − 2
100 = 2𝑛
50 = 𝑛
74
In a classroom of 40 students, each student counts off
by fours (i.e. 4, 8, 12, 16, …). What is the sum of the
students’ numbers?
a1 = 4 d =4
Illustrative Example
Here, n=40
𝑆40 =
𝑛
2
(𝑎1 + 𝑎 𝑛)
𝑆40 =
40
2
(4 + 160)
𝑆40 = 20 (164)
𝑆40 = 𝟑, 𝟐𝟖𝟎
𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑
𝑎 𝑛 = 4 + 40 − 1 4
𝑎 𝑛 = 4 + 156
𝑎 𝑛 = 160
75
76
77
Practice Exercises
Determine if the given sequence is an arithmetic
sequence or not.
1. 6, 8, 10, 12, …
2. -11, -10, -9, -8, …
3. 8, 11, 14, 17, …
4. 5, 15, 45, 135, …
5. 6, 11, 17, 21, …
Arithmetic Sequence NOT
Arithmetic Sequence
Arithmetic Sequence
Arithmetic Sequence
Arithmetic Sequence
NOT
NOT
NOT
NOT
Click the figure which corresponds to your answer. 78
Practice Exercises
Solve the following problems.
6. In the arithmetic sequence , 2, 5, 8, 11, …, what is the 30th
term?
7. In the arithmetic sequence 8, 5, 2, -1, …, what is the 15th term?
8. In the arithmetic sequence with 12 as the first term and the
common difference is -3, what is the 17th term?
87
-28
88 89 90
-30 -32 -34
-36 -37 -38 -39
Click the figure which corresponds to your answer. 79
Practice Exercises
Solve the following problems.
9. In the arithmetic sequence 23, 30, 37, 44, …, what is the 14th
term?
10. In the arithmetic sequence 6, 12, 18, …, what is the 29th term?
110
170
112 114 116
172 174 176
Click the figure which corresponds to your answer. 80
Practice Exercises
Find the arithmetic mean of the following numbers.
11) 4 and 16
12) 19 and 35
13) 13 and 25
14) -22 and 8
15)102 and 1002
-6
8 10 12
26 27 28 29
6
15 17 2119
550
-7
552
-8
554
-9
556
Click the figure which corresponds to your answer. 81
Practice Exercises
Solve the following problems.
16. What is the sum of the first 100 positive odd integers?
17. What is the sum of the first 50 positive even integers?
18. What is the sum of the first 30 positive multiples of 8?
3730
11000 12000 13000
2500
3710
2550 2600 2650
3720
10000
3740
Click the figure which corresponds to your answer. 82
Practice Exercises
19. Aris takes a job, starting with an hourly wage of ₱
350.00 and is promised a raise of ₱ 5.00 per hour every
two months for 5 years. At the end of 5 years, what
would be Aris’ hourly wage?
20. Find the sum of all two-digit even natural numbers
₱ 485 ₱ 490 ₱ 495 ₱ 500
2410 2420 2430 2440
Click the figure which corresponds to your answer. 83
96
Assessment
Determine if the sequence is an arithmetic
sequence or not.
1. 9, 11, 13, 15, …
2. 6, 11, 16, 21, …
3. 1, 2, 4, 8, 16, …
4. -4, 2, 8, 14, …
5. 1, 8, 27, 64, …
97
Assessment Solve the following problems.
6. What is the 11th term in the sequence 6, 9, 12, 15,… ?
7. What is the 24th term of the sequence 16, 19, 22, … ?
8. What is the 25th term of the sequence 12, 9, 6, … ?
9. What is the 30th term in the arithmetic sequence with a
first term of 15 and a common difference of 5?
10. What is the 10th term of the arithmetic sequence with
a first term of 75 and a common difference of -8?
98
Assessment
11. Insert the arithmetic mean of 8 and 28.
12. Insert two arithmetic means between 16 and 31.
13. Insert two arithmetic means between 21 and 33.
14. Insert three arithmetic means between 11 and 35.
15. Insert three arithmetic means between 48 and 84.
99
Assessment Solve the following problems.
16. Joan started a new job with an annual salary of ₱ 150
000 in 2007. If she receives a ₱ 12 000 raise each year, how
much will her annual salary be in 2017?
17. A stack of telephone poles has 30 poles in the bottom
row. There are 29 poles in the second row, 28 in the next
row, and so on. How many poles are there in the 26th row?
18. Josh spent ₱ 150 on August 1, ₱ 170 on August 2. ₱ 190
on August 3, and so on. How much did Josh spend on
August 31?
100
Assessment
19. An object is dropped from a jet plane and falls 32
feet during the first second. If during each successive
second, it falls 40 feet more than the distance in the
preceding second, how far does it fall during the
eleventh second?
20. What is the seating capacity of a movie house with
40 rows of seats if there are 25 seats in the first row,
28 seats in the second row, 31 in the third row, and so
on?
101
107
• Acelajado, Maxima J. (2008). New High School Mathematics II Second
Edition. Makati City: Diwa Learning Systems, Inc.
• Callanta, Melvin M., et al. (2015). Mathematics – Grade 10 Learner’s
Module. Pasig City: Department of Education.
• Orines, Fernando B., et al. (2008). Next Century Mathematics
(Intermediate Algebra) Second Edition. Quezon City: Phoenix Publishing
House, Inc.
• Oronce, Orlando A., Mendoza, Marilyn O. (2010). E-Math II. Manila: Rex
Book Store, Inc.
• http://www.google.com.ph
(Some of the pictures used in this presentation were taken from the said site)
• http://www.slideshare.com
(Some of the examples and exercises of arithmetic sequence and arithmetic
series used in this presentation were taken from the said site)
• https://www.youtube.com/watch?v=HlZky0FL6ck
(The video used in this presentation was taken from the said site)
108
109

More Related Content

What's hot

Geometric Series and Finding the Sum of Finite Geometric Sequence
Geometric Series and Finding the Sum of Finite Geometric SequenceGeometric Series and Finding the Sum of Finite Geometric Sequence
Geometric Series and Finding the Sum of Finite Geometric SequenceFree Math Powerpoints
 
Quadratic inequality
Quadratic inequalityQuadratic inequality
Quadratic inequalityBrian Mary
 
Geometric Sequence
Geometric SequenceGeometric Sequence
Geometric SequenceJoey Valdriz
 
Arithmetic Sequence and Series
Arithmetic Sequence and SeriesArithmetic Sequence and Series
Arithmetic Sequence and Seriesitutor
 
Arithmetic Mean & Arithmetic Series
Arithmetic Mean & Arithmetic SeriesArithmetic Mean & Arithmetic Series
Arithmetic Mean & Arithmetic SeriesFranz DC
 
Harmonic sequence and fibonacci 10
Harmonic sequence and fibonacci 10Harmonic sequence and fibonacci 10
Harmonic sequence and fibonacci 10AjayQuines
 
Geometric Sequence
Geometric SequenceGeometric Sequence
Geometric SequenceJoey Valdriz
 
Geometric Sequence by Alma Baja
Geometric Sequence by Alma BajaGeometric Sequence by Alma Baja
Geometric Sequence by Alma BajaNhatz Marticio
 
Arithmetic sequences and arithmetic means
Arithmetic sequences and arithmetic meansArithmetic sequences and arithmetic means
Arithmetic sequences and arithmetic meansDenmar Marasigan
 
Arithmetic and geometric_sequences
Arithmetic and geometric_sequencesArithmetic and geometric_sequences
Arithmetic and geometric_sequencesDreams4school
 
Geometric Sequence and Geometric Mean
Geometric Sequence and Geometric MeanGeometric Sequence and Geometric Mean
Geometric Sequence and Geometric MeanShemm Madrid
 
Arithmetic sequence
Arithmetic sequenceArithmetic sequence
Arithmetic sequencemaricel mas
 
Finding the sum of a geometric sequence
Finding the sum of a geometric sequenceFinding the sum of a geometric sequence
Finding the sum of a geometric sequencemwagner1983
 
13 1 arithmetic and geometric sequences
13 1 arithmetic and geometric sequences13 1 arithmetic and geometric sequences
13 1 arithmetic and geometric sequenceshisema01
 

What's hot (20)

Geometric Series and Finding the Sum of Finite Geometric Sequence
Geometric Series and Finding the Sum of Finite Geometric SequenceGeometric Series and Finding the Sum of Finite Geometric Sequence
Geometric Series and Finding the Sum of Finite Geometric Sequence
 
Geometric Sequence
Geometric SequenceGeometric Sequence
Geometric Sequence
 
Quadratic inequality
Quadratic inequalityQuadratic inequality
Quadratic inequality
 
Geometric Sequence
Geometric SequenceGeometric Sequence
Geometric Sequence
 
Arithmetic Sequence and Series
Arithmetic Sequence and SeriesArithmetic Sequence and Series
Arithmetic Sequence and Series
 
Arcs and Central Angles
Arcs and Central AnglesArcs and Central Angles
Arcs and Central Angles
 
Arithmetic Mean & Arithmetic Series
Arithmetic Mean & Arithmetic SeriesArithmetic Mean & Arithmetic Series
Arithmetic Mean & Arithmetic Series
 
Harmonic sequence and fibonacci 10
Harmonic sequence and fibonacci 10Harmonic sequence and fibonacci 10
Harmonic sequence and fibonacci 10
 
Geometric Sequence
Geometric SequenceGeometric Sequence
Geometric Sequence
 
Harmonic sequence
Harmonic sequenceHarmonic sequence
Harmonic sequence
 
Geometric Sequence by Alma Baja
Geometric Sequence by Alma BajaGeometric Sequence by Alma Baja
Geometric Sequence by Alma Baja
 
Arithmetic sequences and arithmetic means
Arithmetic sequences and arithmetic meansArithmetic sequences and arithmetic means
Arithmetic sequences and arithmetic means
 
Arithmetic Sequence
Arithmetic SequenceArithmetic Sequence
Arithmetic Sequence
 
Arithmetic and geometric_sequences
Arithmetic and geometric_sequencesArithmetic and geometric_sequences
Arithmetic and geometric_sequences
 
Geometric Sequence and Geometric Mean
Geometric Sequence and Geometric MeanGeometric Sequence and Geometric Mean
Geometric Sequence and Geometric Mean
 
Patterns in Sequences
Patterns in SequencesPatterns in Sequences
Patterns in Sequences
 
Arithmetic sequence
Arithmetic sequenceArithmetic sequence
Arithmetic sequence
 
Finding the sum of a geometric sequence
Finding the sum of a geometric sequenceFinding the sum of a geometric sequence
Finding the sum of a geometric sequence
 
Infinite Geometric Series
Infinite Geometric SeriesInfinite Geometric Series
Infinite Geometric Series
 
13 1 arithmetic and geometric sequences
13 1 arithmetic and geometric sequences13 1 arithmetic and geometric sequences
13 1 arithmetic and geometric sequences
 

Viewers also liked

Sequence and series
Sequence and seriesSequence and series
Sequence and seriesviannafaye
 
Sequences finding a rule
Sequences   finding a ruleSequences   finding a rule
Sequences finding a ruleDreams4school
 
Arithmetic sequence
Arithmetic sequenceArithmetic sequence
Arithmetic sequenceLeah Mel
 
Sequences and series
Sequences and seriesSequences and series
Sequences and seriesLeo Crisologo
 
13 sequences and series
13   sequences and series13   sequences and series
13 sequences and seriesKathManarang
 
Solid figures 6th grade power point
Solid figures 6th grade power pointSolid figures 6th grade power point
Solid figures 6th grade power pointPaula Ortega
 
Geometric Shapes-PowerPoint Slide Show
Geometric Shapes-PowerPoint Slide ShowGeometric Shapes-PowerPoint Slide Show
Geometric Shapes-PowerPoint Slide ShowAlexandra Diana
 
Geometry presentation
Geometry presentationGeometry presentation
Geometry presentationBilly
 

Viewers also liked (12)

Sequence and series
Sequence and seriesSequence and series
Sequence and series
 
Sequences finding a rule
Sequences   finding a ruleSequences   finding a rule
Sequences finding a rule
 
Nth term
Nth termNth term
Nth term
 
Arithmetic sequence
Arithmetic sequenceArithmetic sequence
Arithmetic sequence
 
Sequences and series
Sequences and seriesSequences and series
Sequences and series
 
13 sequences and series
13   sequences and series13   sequences and series
13 sequences and series
 
Solid geometry
Solid geometrySolid geometry
Solid geometry
 
Solid figures 6th grade power point
Solid figures 6th grade power pointSolid figures 6th grade power point
Solid figures 6th grade power point
 
Shapes.ppt
Shapes.pptShapes.ppt
Shapes.ppt
 
Geometric Shapes-PowerPoint Slide Show
Geometric Shapes-PowerPoint Slide ShowGeometric Shapes-PowerPoint Slide Show
Geometric Shapes-PowerPoint Slide Show
 
Properties of 3 d shapes
Properties of 3 d shapesProperties of 3 d shapes
Properties of 3 d shapes
 
Geometry presentation
Geometry presentationGeometry presentation
Geometry presentation
 

Similar to Arithmetic Sequence and Series: Terms, Patterns, and Sums

Generating Patterns and arithmetic sequence.pptx
Generating Patterns and arithmetic sequence.pptxGenerating Patterns and arithmetic sequence.pptx
Generating Patterns and arithmetic sequence.pptxRenoLope1
 
Arithmetic Sequence
Arithmetic SequenceArithmetic Sequence
Arithmetic SequenceJoey Valdriz
 
geometricsequencesandgeometricmeans-150222031045-conversion-gate01.pdf
geometricsequencesandgeometricmeans-150222031045-conversion-gate01.pdfgeometricsequencesandgeometricmeans-150222031045-conversion-gate01.pdf
geometricsequencesandgeometricmeans-150222031045-conversion-gate01.pdfJosephSPalileoJr
 
Successful Minds,Making Mathematics number patterns &sequences Simple.
Successful Minds,Making Mathematics number patterns &sequences Simple.Successful Minds,Making Mathematics number patterns &sequences Simple.
Successful Minds,Making Mathematics number patterns &sequences Simple.Thato Barry
 
Grade 10 Math Module 1 searching for patterns, sequence and series
Grade 10 Math Module 1   searching for patterns, sequence and seriesGrade 10 Math Module 1   searching for patterns, sequence and series
Grade 10 Math Module 1 searching for patterns, sequence and seriesJocel Sagario
 
Week 2: Arithmetic sequence
Week 2:  Arithmetic sequenceWeek 2:  Arithmetic sequence
Week 2: Arithmetic sequenceRozzel Palacio
 
arithmetic sequence.pptx
arithmetic sequence.pptxarithmetic sequence.pptx
arithmetic sequence.pptxCeiCei2
 
090799768954
090799768954090799768954
090799768954FERNAN85
 
Patterns, sequences and series
Patterns, sequences and seriesPatterns, sequences and series
Patterns, sequences and seriesVukile Xhego
 
Patterns & Arithmetic Sequences.pptx
Patterns & Arithmetic Sequences.pptxPatterns & Arithmetic Sequences.pptx
Patterns & Arithmetic Sequences.pptxDeanAriolaSan
 
Arithmetic Sequence.pptx
Arithmetic Sequence.pptxArithmetic Sequence.pptx
Arithmetic Sequence.pptxZaintHarbiHabal
 
Airthmatic sequences with examples
Airthmatic  sequences with  examplesAirthmatic  sequences with  examples
Airthmatic sequences with examplesyousafzufiqar
 
Sequences and Series (S&S GAME) - Barisan dan Deret.pdf
Sequences and Series (S&S GAME) - Barisan dan Deret.pdfSequences and Series (S&S GAME) - Barisan dan Deret.pdf
Sequences and Series (S&S GAME) - Barisan dan Deret.pdfDiah Lutfiana Dewi
 

Similar to Arithmetic Sequence and Series: Terms, Patterns, and Sums (20)

Generating Patterns and arithmetic sequence.pptx
Generating Patterns and arithmetic sequence.pptxGenerating Patterns and arithmetic sequence.pptx
Generating Patterns and arithmetic sequence.pptx
 
Arithmetic Sequence
Arithmetic SequenceArithmetic Sequence
Arithmetic Sequence
 
Yr7-Sequences.pptx
Yr7-Sequences.pptxYr7-Sequences.pptx
Yr7-Sequences.pptx
 
geometricsequencesandgeometricmeans-150222031045-conversion-gate01.pdf
geometricsequencesandgeometricmeans-150222031045-conversion-gate01.pdfgeometricsequencesandgeometricmeans-150222031045-conversion-gate01.pdf
geometricsequencesandgeometricmeans-150222031045-conversion-gate01.pdf
 
Ebook 1
Ebook 1Ebook 1
Ebook 1
 
Successful Minds,Making Mathematics number patterns &sequences Simple.
Successful Minds,Making Mathematics number patterns &sequences Simple.Successful Minds,Making Mathematics number patterns &sequences Simple.
Successful Minds,Making Mathematics number patterns &sequences Simple.
 
Grade 10 Math Module 1 searching for patterns, sequence and series
Grade 10 Math Module 1   searching for patterns, sequence and seriesGrade 10 Math Module 1   searching for patterns, sequence and series
Grade 10 Math Module 1 searching for patterns, sequence and series
 
Week 2: Arithmetic sequence
Week 2:  Arithmetic sequenceWeek 2:  Arithmetic sequence
Week 2: Arithmetic sequence
 
arithmetic sequence.pptx
arithmetic sequence.pptxarithmetic sequence.pptx
arithmetic sequence.pptx
 
090799768954
090799768954090799768954
090799768954
 
Sequence.pptx
Sequence.pptxSequence.pptx
Sequence.pptx
 
Patterns, sequences and series
Patterns, sequences and seriesPatterns, sequences and series
Patterns, sequences and series
 
Patterns & Arithmetic Sequences.pptx
Patterns & Arithmetic Sequences.pptxPatterns & Arithmetic Sequences.pptx
Patterns & Arithmetic Sequences.pptx
 
Sequences.pptx
Sequences.pptxSequences.pptx
Sequences.pptx
 
Sequence function
Sequence functionSequence function
Sequence function
 
ME Math 10 Q1 0104 PS.pptx
ME Math 10 Q1 0104 PS.pptxME Math 10 Q1 0104 PS.pptx
ME Math 10 Q1 0104 PS.pptx
 
Arithmetic Sequence.pptx
Arithmetic Sequence.pptxArithmetic Sequence.pptx
Arithmetic Sequence.pptx
 
Airthmatic sequences with examples
Airthmatic  sequences with  examplesAirthmatic  sequences with  examples
Airthmatic sequences with examples
 
Arithmetic sequences and series
Arithmetic sequences and seriesArithmetic sequences and series
Arithmetic sequences and series
 
Sequences and Series (S&S GAME) - Barisan dan Deret.pdf
Sequences and Series (S&S GAME) - Barisan dan Deret.pdfSequences and Series (S&S GAME) - Barisan dan Deret.pdf
Sequences and Series (S&S GAME) - Barisan dan Deret.pdf
 

More from Joey Valdriz

RA 9155 or Governance of Basic Education Act of 2001
RA 9155 or Governance of Basic Education Act of 2001RA 9155 or Governance of Basic Education Act of 2001
RA 9155 or Governance of Basic Education Act of 2001Joey Valdriz
 
A Presentation on the Learning Action Cell
A Presentation on the Learning Action CellA Presentation on the Learning Action Cell
A Presentation on the Learning Action CellJoey Valdriz
 
A Presentation on the No Collection Policy of DepEd
A Presentation on the No Collection Policy of DepEdA Presentation on the No Collection Policy of DepEd
A Presentation on the No Collection Policy of DepEdJoey Valdriz
 
General Elements of Self-Learning Modules
General Elements of Self-Learning ModulesGeneral Elements of Self-Learning Modules
General Elements of Self-Learning ModulesJoey Valdriz
 
Beyond Hard Skills: Math as Social and Emotional Learning
Beyond Hard Skills: Math as Social and Emotional LearningBeyond Hard Skills: Math as Social and Emotional Learning
Beyond Hard Skills: Math as Social and Emotional LearningJoey Valdriz
 
Problems Involving Probabilities of Events (Math 8)
Problems Involving Probabilities of Events (Math 8)Problems Involving Probabilities of Events (Math 8)
Problems Involving Probabilities of Events (Math 8)Joey Valdriz
 
Conditional Probability
Conditional ProbabilityConditional Probability
Conditional ProbabilityJoey Valdriz
 
Multiplying Polynomials: Two Binomials
Multiplying Polynomials: Two BinomialsMultiplying Polynomials: Two Binomials
Multiplying Polynomials: Two BinomialsJoey Valdriz
 
Summative Test on Measures of Position
Summative Test on Measures of PositionSummative Test on Measures of Position
Summative Test on Measures of PositionJoey Valdriz
 
Probability of Simple and Compound Events
Probability of Simple and Compound EventsProbability of Simple and Compound Events
Probability of Simple and Compound EventsJoey Valdriz
 
Plotting of Points on the Coordinate Plane
Plotting of Points on the Coordinate PlanePlotting of Points on the Coordinate Plane
Plotting of Points on the Coordinate PlaneJoey Valdriz
 
Language Strategies in Teaching Mathematics
Language Strategies in Teaching MathematicsLanguage Strategies in Teaching Mathematics
Language Strategies in Teaching MathematicsJoey Valdriz
 
Philippine National Heroes
Philippine National HeroesPhilippine National Heroes
Philippine National HeroesJoey Valdriz
 
Solving Systems of Linear Equations in Two Variables by Graphing
Solving Systems of Linear Equations in Two Variables by GraphingSolving Systems of Linear Equations in Two Variables by Graphing
Solving Systems of Linear Equations in Two Variables by GraphingJoey Valdriz
 
Interdisciplinary Contextualization for Mathematics Education
Interdisciplinary Contextualization for Mathematics EducationInterdisciplinary Contextualization for Mathematics Education
Interdisciplinary Contextualization for Mathematics EducationJoey Valdriz
 
Contextualization in the Philippines
Contextualization in the PhilippinesContextualization in the Philippines
Contextualization in the PhilippinesJoey Valdriz
 

More from Joey Valdriz (20)

RA 9155 or Governance of Basic Education Act of 2001
RA 9155 or Governance of Basic Education Act of 2001RA 9155 or Governance of Basic Education Act of 2001
RA 9155 or Governance of Basic Education Act of 2001
 
A Presentation on the Learning Action Cell
A Presentation on the Learning Action CellA Presentation on the Learning Action Cell
A Presentation on the Learning Action Cell
 
A Presentation on the No Collection Policy of DepEd
A Presentation on the No Collection Policy of DepEdA Presentation on the No Collection Policy of DepEd
A Presentation on the No Collection Policy of DepEd
 
Probability
ProbabilityProbability
Probability
 
Combination
CombinationCombination
Combination
 
Permutation
PermutationPermutation
Permutation
 
General Elements of Self-Learning Modules
General Elements of Self-Learning ModulesGeneral Elements of Self-Learning Modules
General Elements of Self-Learning Modules
 
Beyond Hard Skills: Math as Social and Emotional Learning
Beyond Hard Skills: Math as Social and Emotional LearningBeyond Hard Skills: Math as Social and Emotional Learning
Beyond Hard Skills: Math as Social and Emotional Learning
 
Design Thinking
Design ThinkingDesign Thinking
Design Thinking
 
Problems Involving Probabilities of Events (Math 8)
Problems Involving Probabilities of Events (Math 8)Problems Involving Probabilities of Events (Math 8)
Problems Involving Probabilities of Events (Math 8)
 
Conditional Probability
Conditional ProbabilityConditional Probability
Conditional Probability
 
Multiplying Polynomials: Two Binomials
Multiplying Polynomials: Two BinomialsMultiplying Polynomials: Two Binomials
Multiplying Polynomials: Two Binomials
 
Summative Test on Measures of Position
Summative Test on Measures of PositionSummative Test on Measures of Position
Summative Test on Measures of Position
 
Probability of Simple and Compound Events
Probability of Simple and Compound EventsProbability of Simple and Compound Events
Probability of Simple and Compound Events
 
Plotting of Points on the Coordinate Plane
Plotting of Points on the Coordinate PlanePlotting of Points on the Coordinate Plane
Plotting of Points on the Coordinate Plane
 
Language Strategies in Teaching Mathematics
Language Strategies in Teaching MathematicsLanguage Strategies in Teaching Mathematics
Language Strategies in Teaching Mathematics
 
Philippine National Heroes
Philippine National HeroesPhilippine National Heroes
Philippine National Heroes
 
Solving Systems of Linear Equations in Two Variables by Graphing
Solving Systems of Linear Equations in Two Variables by GraphingSolving Systems of Linear Equations in Two Variables by Graphing
Solving Systems of Linear Equations in Two Variables by Graphing
 
Interdisciplinary Contextualization for Mathematics Education
Interdisciplinary Contextualization for Mathematics EducationInterdisciplinary Contextualization for Mathematics Education
Interdisciplinary Contextualization for Mathematics Education
 
Contextualization in the Philippines
Contextualization in the PhilippinesContextualization in the Philippines
Contextualization in the Philippines
 

Recently uploaded

Incoming and Outgoing Shipments in 3 STEPS Using Odoo 17
Incoming and Outgoing Shipments in 3 STEPS Using Odoo 17Incoming and Outgoing Shipments in 3 STEPS Using Odoo 17
Incoming and Outgoing Shipments in 3 STEPS Using Odoo 17Celine George
 
Congestive Cardiac Failure..presentation
Congestive Cardiac Failure..presentationCongestive Cardiac Failure..presentation
Congestive Cardiac Failure..presentationdeepaannamalai16
 
Multi Domain Alias In the Odoo 17 ERP Module
Multi Domain Alias In the Odoo 17 ERP ModuleMulti Domain Alias In the Odoo 17 ERP Module
Multi Domain Alias In the Odoo 17 ERP ModuleCeline George
 
Q4-PPT-Music9_Lesson-1-Romantic-Opera.pptx
Q4-PPT-Music9_Lesson-1-Romantic-Opera.pptxQ4-PPT-Music9_Lesson-1-Romantic-Opera.pptx
Q4-PPT-Music9_Lesson-1-Romantic-Opera.pptxlancelewisportillo
 
ROLES IN A STAGE PRODUCTION in arts.pptx
ROLES IN A STAGE PRODUCTION in arts.pptxROLES IN A STAGE PRODUCTION in arts.pptx
ROLES IN A STAGE PRODUCTION in arts.pptxVanesaIglesias10
 
ClimART Action | eTwinning Project
ClimART Action    |    eTwinning ProjectClimART Action    |    eTwinning Project
ClimART Action | eTwinning Projectjordimapav
 
Transaction Management in Database Management System
Transaction Management in Database Management SystemTransaction Management in Database Management System
Transaction Management in Database Management SystemChristalin Nelson
 
week 1 cookery 8 fourth - quarter .pptx
week 1 cookery 8  fourth  -  quarter .pptxweek 1 cookery 8  fourth  -  quarter .pptx
week 1 cookery 8 fourth - quarter .pptxJonalynLegaspi2
 
Grade 9 Quarter 4 Dll Grade 9 Quarter 4 DLL.pdf
Grade 9 Quarter 4 Dll Grade 9 Quarter 4 DLL.pdfGrade 9 Quarter 4 Dll Grade 9 Quarter 4 DLL.pdf
Grade 9 Quarter 4 Dll Grade 9 Quarter 4 DLL.pdfJemuel Francisco
 
Production of Monoclonal Antibodies by Hybridoma Technology.pptx
Production of Monoclonal Antibodies by Hybridoma Technology.pptxProduction of Monoclonal Antibodies by Hybridoma Technology.pptx
Production of Monoclonal Antibodies by Hybridoma Technology.pptxAnupkumar Sharma
 
Active Learning Strategies (in short ALS).pdf
Active Learning Strategies (in short ALS).pdfActive Learning Strategies (in short ALS).pdf
Active Learning Strategies (in short ALS).pdfPatidar M
 
THEORIES OF ORGANIZATION-PUBLIC ADMINISTRATION
THEORIES OF ORGANIZATION-PUBLIC ADMINISTRATIONTHEORIES OF ORGANIZATION-PUBLIC ADMINISTRATION
THEORIES OF ORGANIZATION-PUBLIC ADMINISTRATIONHumphrey A Beña
 
How to Manage Engineering to Order in Odoo 17
How to Manage Engineering to Order in Odoo 17How to Manage Engineering to Order in Odoo 17
How to Manage Engineering to Order in Odoo 17Celine George
 
ICS2208 Lecture6 Notes for SL spaces.pdf
ICS2208 Lecture6 Notes for SL spaces.pdfICS2208 Lecture6 Notes for SL spaces.pdf
ICS2208 Lecture6 Notes for SL spaces.pdfVanessa Camilleri
 
MECHANISMS OF DIFFERENT TYPES OF HYPERSENITIVITY REACTIONS.pptx
MECHANISMS OF DIFFERENT TYPES OF HYPERSENITIVITY REACTIONS.pptxMECHANISMS OF DIFFERENT TYPES OF HYPERSENITIVITY REACTIONS.pptx
MECHANISMS OF DIFFERENT TYPES OF HYPERSENITIVITY REACTIONS.pptxAnupkumar Sharma
 
4.16.24 21st Century Movements for Black Lives.pptx
4.16.24 21st Century Movements for Black Lives.pptx4.16.24 21st Century Movements for Black Lives.pptx
4.16.24 21st Century Movements for Black Lives.pptxmary850239
 
Choosing the Right CBSE School A Comprehensive Guide for Parents
Choosing the Right CBSE School A Comprehensive Guide for ParentsChoosing the Right CBSE School A Comprehensive Guide for Parents
Choosing the Right CBSE School A Comprehensive Guide for Parentsnavabharathschool99
 
MULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptx
MULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptxMULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptx
MULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptxAnupkumar Sharma
 

Recently uploaded (20)

Incoming and Outgoing Shipments in 3 STEPS Using Odoo 17
Incoming and Outgoing Shipments in 3 STEPS Using Odoo 17Incoming and Outgoing Shipments in 3 STEPS Using Odoo 17
Incoming and Outgoing Shipments in 3 STEPS Using Odoo 17
 
Congestive Cardiac Failure..presentation
Congestive Cardiac Failure..presentationCongestive Cardiac Failure..presentation
Congestive Cardiac Failure..presentation
 
FINALS_OF_LEFT_ON_C'N_EL_DORADO_2024.pptx
FINALS_OF_LEFT_ON_C'N_EL_DORADO_2024.pptxFINALS_OF_LEFT_ON_C'N_EL_DORADO_2024.pptx
FINALS_OF_LEFT_ON_C'N_EL_DORADO_2024.pptx
 
Multi Domain Alias In the Odoo 17 ERP Module
Multi Domain Alias In the Odoo 17 ERP ModuleMulti Domain Alias In the Odoo 17 ERP Module
Multi Domain Alias In the Odoo 17 ERP Module
 
Q4-PPT-Music9_Lesson-1-Romantic-Opera.pptx
Q4-PPT-Music9_Lesson-1-Romantic-Opera.pptxQ4-PPT-Music9_Lesson-1-Romantic-Opera.pptx
Q4-PPT-Music9_Lesson-1-Romantic-Opera.pptx
 
ROLES IN A STAGE PRODUCTION in arts.pptx
ROLES IN A STAGE PRODUCTION in arts.pptxROLES IN A STAGE PRODUCTION in arts.pptx
ROLES IN A STAGE PRODUCTION in arts.pptx
 
ClimART Action | eTwinning Project
ClimART Action    |    eTwinning ProjectClimART Action    |    eTwinning Project
ClimART Action | eTwinning Project
 
Transaction Management in Database Management System
Transaction Management in Database Management SystemTransaction Management in Database Management System
Transaction Management in Database Management System
 
Mattingly "AI & Prompt Design: Large Language Models"
Mattingly "AI & Prompt Design: Large Language Models"Mattingly "AI & Prompt Design: Large Language Models"
Mattingly "AI & Prompt Design: Large Language Models"
 
week 1 cookery 8 fourth - quarter .pptx
week 1 cookery 8  fourth  -  quarter .pptxweek 1 cookery 8  fourth  -  quarter .pptx
week 1 cookery 8 fourth - quarter .pptx
 
Grade 9 Quarter 4 Dll Grade 9 Quarter 4 DLL.pdf
Grade 9 Quarter 4 Dll Grade 9 Quarter 4 DLL.pdfGrade 9 Quarter 4 Dll Grade 9 Quarter 4 DLL.pdf
Grade 9 Quarter 4 Dll Grade 9 Quarter 4 DLL.pdf
 
Production of Monoclonal Antibodies by Hybridoma Technology.pptx
Production of Monoclonal Antibodies by Hybridoma Technology.pptxProduction of Monoclonal Antibodies by Hybridoma Technology.pptx
Production of Monoclonal Antibodies by Hybridoma Technology.pptx
 
Active Learning Strategies (in short ALS).pdf
Active Learning Strategies (in short ALS).pdfActive Learning Strategies (in short ALS).pdf
Active Learning Strategies (in short ALS).pdf
 
THEORIES OF ORGANIZATION-PUBLIC ADMINISTRATION
THEORIES OF ORGANIZATION-PUBLIC ADMINISTRATIONTHEORIES OF ORGANIZATION-PUBLIC ADMINISTRATION
THEORIES OF ORGANIZATION-PUBLIC ADMINISTRATION
 
How to Manage Engineering to Order in Odoo 17
How to Manage Engineering to Order in Odoo 17How to Manage Engineering to Order in Odoo 17
How to Manage Engineering to Order in Odoo 17
 
ICS2208 Lecture6 Notes for SL spaces.pdf
ICS2208 Lecture6 Notes for SL spaces.pdfICS2208 Lecture6 Notes for SL spaces.pdf
ICS2208 Lecture6 Notes for SL spaces.pdf
 
MECHANISMS OF DIFFERENT TYPES OF HYPERSENITIVITY REACTIONS.pptx
MECHANISMS OF DIFFERENT TYPES OF HYPERSENITIVITY REACTIONS.pptxMECHANISMS OF DIFFERENT TYPES OF HYPERSENITIVITY REACTIONS.pptx
MECHANISMS OF DIFFERENT TYPES OF HYPERSENITIVITY REACTIONS.pptx
 
4.16.24 21st Century Movements for Black Lives.pptx
4.16.24 21st Century Movements for Black Lives.pptx4.16.24 21st Century Movements for Black Lives.pptx
4.16.24 21st Century Movements for Black Lives.pptx
 
Choosing the Right CBSE School A Comprehensive Guide for Parents
Choosing the Right CBSE School A Comprehensive Guide for ParentsChoosing the Right CBSE School A Comprehensive Guide for Parents
Choosing the Right CBSE School A Comprehensive Guide for Parents
 
MULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptx
MULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptxMULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptx
MULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptx
 

Arithmetic Sequence and Series: Terms, Patterns, and Sums

  • 2. 2
  • 3. At the end of the lesson, the learners are expected to: 1) Generate and describe patterns using symbols and mathematical expressions; 2) Find the next few terms and the nth term of the given sequence; 3) Define and describe an arithmetic sequence; 4) Enumerate the next few terms and the nth term of an arithmetic sequence; 5) Insert means between two given terms of an arithmetic sequence; 6) Find the sum of the first n terms of an arithmetic sequence; and 7) Solve problems involving arithmetic sequence. 3
  • 4. 4
  • 5. 5
  • 6. 6
  • 7. 1. O T T F F S S E N _? T E F S 2. R O Y B G I _? T U V W 3. J F M A M J J A S O N _? A B C D 4. T Q P H H O N _? D U H I Direction: Click the letter of your answer. 7
  • 8. 11
  • 9. 12
  • 10. 1) 80, 40, 20, 10, 5, __? 2) 1, 8, 27, 64, 125, 216, __? 3) 1, 4, 9, 16, 25, 36, 49, __? 343 512 729 810 56 64 72 81 0 1 2 1 5 2 13
  • 11. 4) 1, 2, 3, 5, 8, 13, 21, __? 5) -2, -5, -8, -11, -14, __? 6) 7, 11, 15, 19, 23, 27, __? -17 -19 -21 -23 29 30 31 32 31 32 33 34 14
  • 12. 19
  • 13. 20
  • 14. 21
  • 15. 22
  • 16. 23
  • 17. 24
  • 18. 25
  • 19. 26
  • 20. 27
  • 21. 28
  • 22. 29
  • 23. 30
  • 24. 31
  • 25. An ordered list of numbers, such as 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, … is called a sequence. A number sequence is a list of numbers having a first number, a second number, a third number, and so on, called the terms of the sequence. 32
  • 26. FINITE AND INFINITE SEQUENCES • 10, 15, 20, 25, 30 • There are only a finite number of terms. • They have a last term. • 11, 22, 33, 44, 55, … • There are infinite number of terms. • They do not have a last term. 33
  • 27. Sequences are usually given by stating their general or nth term. NUMBER SEQUENCE Example: Consider the sequence given 𝒂 𝒏 = 𝟑𝒏 + 𝟐 The first five terms of the sequence are 𝒂 𝟏 = 𝟑 𝟏 + 𝟐 = 𝟑 + 𝟐 = 𝟓, 𝒂 𝟐 = 𝟑 𝟐 + 𝟐 = 𝟔 + 𝟐 = 𝟖, 𝒂 𝟑 = 𝟑 𝟑 + 𝟐 = 𝟗 + 𝟐 = 𝟏𝟏, 𝒂 𝟒 = 𝟑 𝟒 + 𝟐 = 𝟏𝟐 + 𝟐 = 𝟏𝟒, 𝒂𝒏𝒅 𝒂 𝟓 = 𝟑 𝟓 + 𝟐 = 𝟏𝟓 + 𝟐 = 𝟏𝟕 34
  • 28. a. 𝒂 𝒏 = 𝟐𝒏 𝟐 for 𝟏 ≤ 𝒏 ≤ 𝟓 b. 𝒂 𝒏 = 𝟐𝒏 − 𝟏 for 𝟏 ≤ 𝒏 ≤ 𝟓 c. 𝒂 𝒏 = (−𝟏) 𝒏 (𝒏 − 𝟑) 𝟐 for 𝟏 ≤ 𝒏 ≤ 𝟒 List all the indicated terms of each finite sequence. NUMBER SEQUENCE 35
  • 29. 37
  • 30. QUESTION: What do you observe in the following pictures 38
  • 31. 39
  • 32. Joey’s school offered him a scholarship grant of ₱ 3,000.00 when he was in Grade 7 and increased the amount by ₱ 500 each year till Grade 10. The amounts of money (in ₱) Joey received in Grade 7, 8, 9, and 10 were respectively: 3000, 3500, 4000, and 4500 Each of the numbers in the list is called a term. Note: We find that the succeeding terms are obtained by adding a fixed number. 40
  • 33. In a savings scheme, the amount triples after every 7 years. The maturity amount (in ₱) of an investment of ₱ 6,000.00 after 7, 14, 21 and 28 years will be, respectively: 18000, 54000, 162000, 486000 Note: We find that the succeeding terms are obtained by multiplying with a fixed number. 41
  • 34. NUMBER PATTERNS The number of unit squares in a square with sides 1, 2, 3, 4, ... units are respectively 1, 4, 9, 16, .... Note: We can observe that 1=12, 4=22, 9=32, 16=42, ... Thus, the succeeding terms are squares of consecutive numbers. 42
  • 35. 43
  • 36. ARITHMETIC SEQUENCE Consider the following lists of numbers : 2, 4, 6, 8, 10, .... 15, 12, 9, 6, 3, .... -5, -4,-3, -2, -1.... 6, 6, 6, 6, 6, 6, .... each term is obtained by adding 2 to the previous term each term is obtained by adding -3 to the previous term each term is obtained by adding 1 to the previous term each term is obtained by adding 0 to the previous term 44
  • 37. 45
  • 38. 1) 23, 38, 53, __ , 83, 98 2) 45, 37, __ , 21, 13, 5 3) -13, -6, __ , 8, 15, 22 27 28 29 30 0 1 -2 -3 63 68 73 78 46
  • 39. 4) __ , 23, 32, 41, 50, 59 5) -12, -7, -2, 3, 8, ___ 6) 10, __ , 32, 43, 54, 65 10 11 12 13 21 23 25 27 10 12 14 16 47
  • 40. The constant is called the common difference, denoted with the letter d, referring to the fact that the difference between two successive terms yields the constant value that was added. To find the common difference, subtract the first term from the second term. In mathematics, an arithmetic sequence or arithmetic progression is a sequence of numbers where each term after the first term is obtained by adding the same constant (always the same). 52
  • 41. Examples: 5, 10, 15, 20, 25, … 3, 7, 11, 15, … 20, 18, 16, 14, … -7, -17, -27, -37, … d=5 d=4 d=-2 d=-10 53
  • 42. 2, 5, 8, 11, 14 … In this sequence, notice that the common difference is 3. In order to get to the next term in the sequence, we will add 3; so, a recursive formula for this sequence is: 31  nn aa The first term in the sequence is a1 (sometimes just written as a). Example: 54
  • 43. 2, 5, 8, 11, 14 … +3 +3 +3 +3 Every time you want another term in the said sequence of numbers, you have to add d. This means that the second term was the first term plus d. The third term is the first term plus d plus d (added twice). The fourth term is the first term plus d plus d plus d (added three times). So, you can see to get the nth term we have to take the first term and add (n - 1) times d. 𝑑 = 3  dnaan 1 𝑎 = 2   1715231626 a 55
  • 44. To find any term of an arithmetic sequence: where: a1 is the first term of the sequence, d is the common difference, n is the number of the term to find. ARITHMETIC SEQUENCE 56
  • 45. Find the nth term of the arithmetic sequence 5, 8, 11, 14, … We know that 𝒂 𝟏 = 𝟓 𝒅 = 𝟑 Substituting in the formula, we obtain 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 𝑎 𝑛 = 5 + 𝑛 − 1 3 𝑎 𝑛 = 5 + 3𝑛 − 3 𝒂 𝒏 = 𝟐 + 𝟑𝒏 57
  • 46. Find the 18th term of the arithmetic sequence 21, 24, 27, 30, 33, … Note that 𝒂 𝟏 = 𝟓, 𝒅 = 𝟑, 𝒏 = 𝟏𝟖. Using the formula for the general term of an arithmetic sequence, we have 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 𝑎 𝑛 = 21 + 18 − 1 3 𝑎 𝑛 = 21 + (17)3 𝒂 𝒏 = 𝟕𝟐 Thus, 72 is the 18th term of the sequence. 58
  • 47. In the arithmetic sequence 10, 16, 22, 28, 34, … , which term is 124? The problem asks for n when 𝑎 𝑛 = 124. From the given sequence 𝒂 𝟏 = 𝟏𝟎, 𝒅 = 𝟔, 𝒂 𝒏 = 𝟏𝟐𝟒. Substitute these values in the formula to get 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 124 = 10 + 𝑛 − 1 6 124 = 10 + 6𝑛 − 6 120 = 6𝑛 𝟐𝟎 = 𝒏 Thus, 124 is the 20th term. 59
  • 48. Find the first term of the arithmetic sequence with a common difference of 6 and whose 20th term is 965? Since 𝒂 𝟐𝟎 = 𝟗𝟔𝟓, 𝒅 = 𝟔, 𝒏 = 20 , we have 𝑎20 = 𝑎1 + 𝑛 − 1 𝑑 965 = 𝑎1 + 20 − 1 6 965 = 𝑎1 + 19 6 965 = 𝑎1 + 114 𝑎1 = 965 − 114 = 851 Thus, 851 is the first term. 60
  • 49. Find the 15th term of the arithmetic sequence whose first term is 7 and whose 5th term is 19. Find the common difference, d by substituting 𝑎1 = 7 and, 𝑎5 = 19 . Thus, 𝑎5 = 𝑎1 + 𝑛 − 1 𝑑 19 = 7 + 5 − 1 𝑑 19 = 7 + 4𝑑 12 = 4𝑑 3 = 𝑑 Now use 𝒂 𝟏 = 𝟕, 𝒅 = 𝟑, 𝒏 = 𝟏𝟓 𝒂 𝟏𝟓 = 𝟕 + 𝟏𝟓 − 𝟏 𝟑 = 𝟒𝟗 Thus, the 15th term is 49. 61
  • 50. Insert two arithmetic means between 8 and 17. Our task here is to find 𝑎2 and 𝑎3 such that 8, 𝑎2, 𝑎3, 17 form an arithmetic sequence. 𝑎1 = 8 𝑎4 = 17 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 17 = 8 + 4 − 1 𝑑 17 = 8 + 3𝑑 15 = 3𝑑 5 = 𝑑 We need 𝑎2 and 𝑎3 . Hence, 𝑎2 = 8 + 3 = 𝟏𝟏 and 𝑎3 = 8 + 3 + 3 = 𝟏𝟒 62
  • 51. A conference hall has 33 rows of seats. The last row contains 80 seats. If each row has two fewer seats than the row behind it. How many seats are there in the first row? We know that 𝒂 𝟑𝟑 = 𝟖𝟎, 𝒅 = 𝟐, 𝒏 = 33 . Using the formula for the general term of an arithmetic sequence, we have 𝑎33 = 𝑎1 + 𝑛 − 1 𝑑 80 = 𝑎1 + 33 − 1 2 80 = 𝑎1 + 32 2 80 = 𝑎1 + 64 𝑎1 = 80 − 64 = 16 Therefore, there are 16 seats in the first row. 63
  • 52. How many numbers between 7 and 500 are divisible by 5? The common difference, d, is 5. Since we want numbers that are divisible by 5, then we let 𝒂 𝟏 = 𝟏𝟎, 𝒂 𝒏 = 𝟓𝟎𝟎. Substitute these values into the general term. 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 500 = 10 + 𝑛 − 1 5 500 = 10 + 5𝑛 − 5 495 = 5𝑛 𝟗𝟗 = 𝒏 There are 99 numbers between 7 and 500 that are divisible by 5. 64
  • 53. 65
  • 54. 1. 1, 2, 3, 4, … and 1+2+3+4+… 2. 2, 4, 6, 8, … and 2+4+6+8+… 3. 5, 10, 15, … and 5+10+15+… 4. 3, 7, 11, 15, … and 3+7+11+15+… 5. 4, 8, 12, 16, … and 4+8+12+16+… Each indicated sum of the terms of an arithmetic sequence is an arithmetic series. 66
  • 55. Do you know this? The story is told of a grade school teacher In the 1700’s that wanted to keep her class busy while she graded papers so she asked them to add up all of the numbers from 1 to 100. These numbers are an arithmetic sequence with common difference 1. Carl Friedrich Gauss was in the class and had the answer in a minute or two (remember no calculators in those days). This is what he did: 1 + 2 + 3 + 4 + 5 + . . . + 96 + 97 + 98 + 99 + 100 sum is 101 sum is 101 67
  • 56. The formula for the sum of n terms is:  nn aa n S  1 2 n is the number of terms so 𝒏 𝟐 would be the number of pairs Let’s find the sum of 1 + 3 +5 + . . . + 59 ARITHMETIC SERIES first term last term 68
  • 57.  nn aa n S  1 2 Let’s find the sum of 1 + 3 +5 + . . . + 59  12 nThe common difference is 2 and the first term is 1, so: Set this equal to 59 to find n. Remember n is the term number. 𝟐𝒏 − 𝟏 = 𝟓𝟗 𝒏 = 𝟑𝟎 So there are 30 terms to sum up.   900591 2 30 30 S first term last term 69
  • 58. To find the sum of a certain number of terms of an arithmetic sequence: where: Sn is the sum of n terms (nth partial sum), a1 is the first term, an is the nth term. 70
  • 59. To find the sum of a certain number of terms of an arithmetic sequence: where: Sn is the sum of n terms (nth partial sum), a is the first term, n is the “rank” of the nth term d is the common difference 71
  • 60. Find the sum of the first ten positive integers. a1 = 1 n = 10 Illustrative Example a10 = 10 𝑆10 = 𝑛 2 (𝑎1 + 𝑎 𝑛) 𝑆10 = 10 2 (1 + 10) 𝑆10 = 5 (11) 𝑆10 = 𝟓𝟓 72
  • 61. Find the sum of the first 15 terms of the arithmetic sequence if the first term is 11 and the 15th term is 109. a1 = 11 n = 15 Illustrative Example a15 = 109 𝑆15 = 𝑛 2 (𝑎1 + 𝑎 𝑛) 𝑆15 = 15 2 (11 + 109) 𝑆15 = 15 2 (120) 𝑆15 = 𝟗𝟎𝟎 73
  • 62. Find the sum of all the odd integers from 1 to 99. a1 = 1 d = 2 Illustrative Example Here, a10 = 99 𝑆50 = 𝑛 2 (𝑎1 + 𝑎 𝑛) 𝑆50 = 50 2 (1 + 99) 𝑆50 = 25 (100) 𝑆50 = 𝟐, 𝟓𝟎𝟎 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 99 = 1 + 𝑛 − 1 2 99 = 1 + 2𝑛 − 2 100 = 2𝑛 50 = 𝑛 74
  • 63. In a classroom of 40 students, each student counts off by fours (i.e. 4, 8, 12, 16, …). What is the sum of the students’ numbers? a1 = 4 d =4 Illustrative Example Here, n=40 𝑆40 = 𝑛 2 (𝑎1 + 𝑎 𝑛) 𝑆40 = 40 2 (4 + 160) 𝑆40 = 20 (164) 𝑆40 = 𝟑, 𝟐𝟖𝟎 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 𝑎 𝑛 = 4 + 40 − 1 4 𝑎 𝑛 = 4 + 156 𝑎 𝑛 = 160 75
  • 64. 76
  • 65. 77
  • 66. Practice Exercises Determine if the given sequence is an arithmetic sequence or not. 1. 6, 8, 10, 12, … 2. -11, -10, -9, -8, … 3. 8, 11, 14, 17, … 4. 5, 15, 45, 135, … 5. 6, 11, 17, 21, … Arithmetic Sequence NOT Arithmetic Sequence Arithmetic Sequence Arithmetic Sequence Arithmetic Sequence NOT NOT NOT NOT Click the figure which corresponds to your answer. 78
  • 67. Practice Exercises Solve the following problems. 6. In the arithmetic sequence , 2, 5, 8, 11, …, what is the 30th term? 7. In the arithmetic sequence 8, 5, 2, -1, …, what is the 15th term? 8. In the arithmetic sequence with 12 as the first term and the common difference is -3, what is the 17th term? 87 -28 88 89 90 -30 -32 -34 -36 -37 -38 -39 Click the figure which corresponds to your answer. 79
  • 68. Practice Exercises Solve the following problems. 9. In the arithmetic sequence 23, 30, 37, 44, …, what is the 14th term? 10. In the arithmetic sequence 6, 12, 18, …, what is the 29th term? 110 170 112 114 116 172 174 176 Click the figure which corresponds to your answer. 80
  • 69. Practice Exercises Find the arithmetic mean of the following numbers. 11) 4 and 16 12) 19 and 35 13) 13 and 25 14) -22 and 8 15)102 and 1002 -6 8 10 12 26 27 28 29 6 15 17 2119 550 -7 552 -8 554 -9 556 Click the figure which corresponds to your answer. 81
  • 70. Practice Exercises Solve the following problems. 16. What is the sum of the first 100 positive odd integers? 17. What is the sum of the first 50 positive even integers? 18. What is the sum of the first 30 positive multiples of 8? 3730 11000 12000 13000 2500 3710 2550 2600 2650 3720 10000 3740 Click the figure which corresponds to your answer. 82
  • 71. Practice Exercises 19. Aris takes a job, starting with an hourly wage of ₱ 350.00 and is promised a raise of ₱ 5.00 per hour every two months for 5 years. At the end of 5 years, what would be Aris’ hourly wage? 20. Find the sum of all two-digit even natural numbers ₱ 485 ₱ 490 ₱ 495 ₱ 500 2410 2420 2430 2440 Click the figure which corresponds to your answer. 83
  • 72. 96
  • 73. Assessment Determine if the sequence is an arithmetic sequence or not. 1. 9, 11, 13, 15, … 2. 6, 11, 16, 21, … 3. 1, 2, 4, 8, 16, … 4. -4, 2, 8, 14, … 5. 1, 8, 27, 64, … 97
  • 74. Assessment Solve the following problems. 6. What is the 11th term in the sequence 6, 9, 12, 15,… ? 7. What is the 24th term of the sequence 16, 19, 22, … ? 8. What is the 25th term of the sequence 12, 9, 6, … ? 9. What is the 30th term in the arithmetic sequence with a first term of 15 and a common difference of 5? 10. What is the 10th term of the arithmetic sequence with a first term of 75 and a common difference of -8? 98
  • 75. Assessment 11. Insert the arithmetic mean of 8 and 28. 12. Insert two arithmetic means between 16 and 31. 13. Insert two arithmetic means between 21 and 33. 14. Insert three arithmetic means between 11 and 35. 15. Insert three arithmetic means between 48 and 84. 99
  • 76. Assessment Solve the following problems. 16. Joan started a new job with an annual salary of ₱ 150 000 in 2007. If she receives a ₱ 12 000 raise each year, how much will her annual salary be in 2017? 17. A stack of telephone poles has 30 poles in the bottom row. There are 29 poles in the second row, 28 in the next row, and so on. How many poles are there in the 26th row? 18. Josh spent ₱ 150 on August 1, ₱ 170 on August 2. ₱ 190 on August 3, and so on. How much did Josh spend on August 31? 100
  • 77. Assessment 19. An object is dropped from a jet plane and falls 32 feet during the first second. If during each successive second, it falls 40 feet more than the distance in the preceding second, how far does it fall during the eleventh second? 20. What is the seating capacity of a movie house with 40 rows of seats if there are 25 seats in the first row, 28 seats in the second row, 31 in the third row, and so on? 101
  • 78. 107
  • 79. • Acelajado, Maxima J. (2008). New High School Mathematics II Second Edition. Makati City: Diwa Learning Systems, Inc. • Callanta, Melvin M., et al. (2015). Mathematics – Grade 10 Learner’s Module. Pasig City: Department of Education. • Orines, Fernando B., et al. (2008). Next Century Mathematics (Intermediate Algebra) Second Edition. Quezon City: Phoenix Publishing House, Inc. • Oronce, Orlando A., Mendoza, Marilyn O. (2010). E-Math II. Manila: Rex Book Store, Inc. • http://www.google.com.ph (Some of the pictures used in this presentation were taken from the said site) • http://www.slideshare.com (Some of the examples and exercises of arithmetic sequence and arithmetic series used in this presentation were taken from the said site) • https://www.youtube.com/watch?v=HlZky0FL6ck (The video used in this presentation was taken from the said site) 108
  • 80. 109