The document discusses arithmetic sequences and series. It provides examples of arithmetic sequences and how to determine the common difference and write terms. It also discusses how to find individual terms, sums of terms, and arithmetic means within sequences. Formulas are provided for the nth term of a sequence, the sum of terms, and evaluating sequences.
* Find zeros of polynomial functions
* Use the Fundamental Theorem of Algebra to find a function that satisfies given conditions
* Find all zeros of a polynomial function
* Find zeros of polynomial functions
* Use the Fundamental Theorem of Algebra to find a function that satisfies given conditions
* Find all zeros of a polynomial function
This learner's module talks about the topic or lesson Proportional Segments. This also includes the types of Proportional Segments and its corresponding definitions and examples of each Proportional Segment.
Contents:
1. Geometric Sequence
2. Geometric Means
3. Geometric Series
with activities
Feel free to send me a message at regie.naungayan@deped.gov.ph for corrections or other suggestions
This learner's module talks about the topic or lesson Proportional Segments. This also includes the types of Proportional Segments and its corresponding definitions and examples of each Proportional Segment.
Contents:
1. Geometric Sequence
2. Geometric Means
3. Geometric Series
with activities
Feel free to send me a message at regie.naungayan@deped.gov.ph for corrections or other suggestions
SEQUENCE AND SERIES
SEQUENCE
Is a set of numbers written in a definite order such that there is a rule by which the terms are obtained. Or
Is a set of number with a simple pattern.
Example
1. A set of even numbers
• 2, 4, 6, 8, 10 ……
2. A set of odd numbers
• 1, 3, 5, 7, 9, 11….
Knowing the pattern the next number from the previous can be obtained.
Example
1. Find the next term from the sequence
• 2, 7, 12, 17, 22, 27, 32
The next term is 37.
2. Given the sequence
• 2, 4, 6, 8, 10, 12………
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
2. An introduction…………
1, 4, 7,10,13
9,1, 7, 15
6.2, 6.6, 7, 7.4
, 3, 6
Arithmetic Sequences
ADD
To get next term
2, 4, 8,16, 32
9, 3,1, 1/3
1,1/ 4,1/16,1/ 64
, 2.5 , 6.25
Geometric Sequences
MULTIPLY
To get next term
Arithmetic Series
Sum of Terms
35
12
27.2
3 9
Geometric Series
Sum of Terms
62
20/3
85/ 64
9.75
3. USING AND WRITING SEQUENCES
The numbers in sequences are called terms.
You can think of a sequence as a function whose domain
is a set of consecutive integers. If a domain is not
specified, it is understood that the domain starts with 1.
4. The domain gives
the relative position
of each term.
1 2 3 4 5
DOMAIN:
3 6 9 12 15
RANGE:
The range gives the
terms of the sequence.
This is a finite sequence having the rule
an = 3n,
where an represents the nth term of the sequence.
USING AND WRITING SEQUENCES
n
an
5. Writing Terms of Sequences
Write the first five terms of the sequence an = 2n + 3.
SOLUTION
a1 = 2(1) + 3 = 5 1st term
2nd term
3rd term
4th term
a2 = 2(2) + 3 = 7
a3 = 2(3) + 3 = 9
a4 = 2(4) + 3 = 11
a5 = 2(5) + 3 = 13 5th term
6. Writing Terms of Sequences
SOLUTION
3rd term
Write the first five terms of the sequence .
1
n
n
an
2
1
1
)
1
(
)
1
(
1
a 1st term
3
2
1
)
2
(
)
2
(
2
a 2nd term
4
3
1
)
3
(
)
3
(
3
a
5
4
1
)
4
(
)
4
(
4
a
6
5
1
)
5
(
)
5
(
5
a
4th term
5th term
7. Writing Terms of Sequences
Write the first five terms of the sequence an = n! - 2.
SOLUTION
a1 = (1)!-2 = -1 1st term
2nd term
3rd term
4th term
a2 = (2)! - 2 = 0
a3 = (3)! - 2 = 4
a4 = (4)! - 2 = 22
a5 = (5)! - 2 = 118 5th term
8. Writing Terms of Sequences
Write the first five terms of the sequence f(n) = (–2)n – 1
.
SOLUTION
f(1) = (–2) 1 – 1
= 1 1st term
2nd term
3rd term
4th term
f(2) = (–2) 2 – 1
= –2
f(3) = (–2) 3 – 1
= 4
f(4) = (–2) 4 – 1
= – 8
f(5) = (–2) 5 – 1
= 16 5th term
9. Arithmetic Sequences and Series
Arithmetic Sequence: sequence whose consecutive terms
have a common difference.
Example: 3, 5, 7, 9, 11, 13, ...
The terms have a common difference of 2.
The common difference is the number d.
To find the common difference you use an+1 – an
Example: Is the sequence arithmetic?
–45, –30, –15, 0, 15, 30
Yes, the common difference is 15
10. Find the next four terms of –9, -2, 5, …
Arithmetic Sequence
7 is referred to as the common difference (d)
Common Difference (d) – what we ADD to get next term
Next four terms……12, 19, 26, 33
-2 - -9 = 7 and 5 - -2 = 7
11. Find the next four terms of 0, 7, 14, …
Arithmetic Sequence, d = 7
21, 28, 35, 42
Find the next four terms of -3x, -2x, -x, …
Arithmetic Sequence, d = x
0, x, 2x, 3x
Find the next four terms of 5k, -k, -7k, …
Arithmetic Sequence, d = -6k
-13k, -19k, -25k, -31k
12. How do you find any term in this sequence?
To find any term in an arithmetic sequence, use the formula
an = a1 + (n – 1)d
where d is the common difference.
13. Vocabulary of Sequences (Universal)
1
a First term
n
a nth term
n
S sum of n terms
n number of terms
d common difference
n 1
n 1 n
nth term of arithmetic sequence
sum of n terms of arithmetic sequen
a a n 1 d
n
S a a
2
ce
14. Find the 14th term of the
arithmetic sequence
4, 7, 10, 13,……
1 ( 1)
n
a a n d
14
a (14 1)
4 3
4 (13)3
4 39
43
15. 16 1
Find a if a 1.5 and d 0.5
Try this one:
1
a First term
n
a nth term
n
S sum of n terms
n number of terms
d common difference
1.5
16
a16
NA
0.5
n 1
a a n 1 d
a16 = 1.5 + (16 - 1)(0.5)
a16 = 1.5 + (15)(0.5)
a16 = 1.5+7.5
a16 = 9
16. Months Cost ($)
1 75,000
2 90,000
3 105,000
4 120,000
The table shows typical costs for a construction company to rent a
crane for one, two, three, or four months. Assuming that the arithmetic
sequence continues, how much would it cost to rent the crane for
twelve months?
75,000
12
a12
NA
15,000
1
a First term
n
a nth term
n
S sum of n terms
n number of terms
d common difference
n 1
a a n 1 d
a12 = 75,000 + (12 - 1)(15,000)
a12 = 75,000 + (11)(15,000)
a12 = 75,000+165,000
a12 = $240,000
17. 1 ( 1)
n
a a n d
In the arithmetic sequence
4,7,10,13,…, which term has a
value of 301?
301 4 ( 1)3
n
301 4 3 3
n
301 1 3n
300 3n
100 n
18. n 1
Findnif a 633, a 9, and d 24
1
a First term
n
a nth term
n
S sum of n terms
n number of terms
d common difference
9
n
633
NA
24
n 1
a a n 1 d
Try this one:
633 = 9 + (n - 1)(24)
633 = 9 + 24n - 24
633 = 24n – 15
648 = 24n
n = 27
19. Given an arithmetic sequence with 15 1
a 38 and d 3, find a .
1
a First term
n
a nth term
n
S sum of n terms
n number of terms
d common difference
a1
15
38
NA
-3
n 1
a a n 1 d
38 = a1 + (15 - 1)(-3)
38 = a1 + (14)(-3)
38 = a1 - 42
a1 = 80
20. 1 29
Find dif a 6 and a 20
-6
29
20
NA
d
1
a First term
n
a nth term
n
S sum of n terms
n number of terms
d common difference
n 1
a a n 1 d
20 = -6 + (29 - 1)(d)
20 = -6 + (28)(d)
26 = 28d
14
13
d
21. Write an equation for the nth term of the arithmetic
sequence 8, 17, 26, 35, …
1
a First term
d common difference
8
9
an = 8 + (n - 1)(9)
an = 8 + 9n - 9
an = 9n - 1
n 1
a a n 1 d
22. Arithmetic Mean: The terms between any two
nonconsecutive terms of an arithmetic sequence.
Ex. 19, 30, 41, 52, 63, 74, 85, 96
41, 52, 63 are the Arithmetic Mean between 30 and 74
23. Find two arithmetic means between –4 and 5
-4, ____, ____, 5
1
a First term
n
a nth term
n
S sum of n terms
n number of terms
d common difference
-4
4
5
NA
d
n 1
a a n 1 d
The two arithmetic means are –1 and 2, since –4, -1, 2, 5
forms an arithmetic sequence
5 = -4 + (4 - 1)(d)
5 = -4 + (3)(d)
9 = (3)(d)
d = 3
24. Find three arithmetic means between 21 and 45
21, ____, ____, ____, 45
1
a First term
n
a nth term
n
S sum of n terms
n number of terms
d common difference
21
5
45
NA
d
n 1
a a n 1 d
The three arithmetic means are 27, 33, and 39
since 21, 27, 33, 39, 45 forms an arithmetic sequence
45 = 21 + (5 - 1)(d)
45 = 21 + (4)(d)
24 = (4)(d)
d = 6
25. Arithmetic Series: An indicated sum of terms in an
arithmetic sequence.
Example:
Arithmetic Sequence
3, 5, 7, 9, 11, 13
VS Arithmetic Series
3 + 5 + 7 + 9 + 11 + 13
26. Vocabulary of Sequences (Universal)
1
a First term
n
a nth term
n
S sum of n terms
n number of terms
d common difference
Recall
27. 1
a First term
n
a nth term
n
S sum of n terms
n number of terms
d common difference
-19
63
??
Sn
6
n 1
a a n 1 d
353
n 1 n
n
S a a
2
63
63
3 3
S
2
19 5
63 1 1
S 052
Find the sum of the first 63 terms of the arithmetic sequence -19, -13, -7,…
n 1 n
n
S a a
2
a63 = 353
28. Find the first 3 terms for an arithmetic series in which a1 = 9, an = 105, Sn =741.
1
a First term
n
a nth term
n
S sum of n terms
n number of terms
d common difference
9
??
105
741
??
n 1
a a n 1 d
n 1 n
n
S a a
2
13
n 1
a a n 1 d
9, 17, 25
29. A radio station considered giving away $4000 every day in the month of August for a
total of $124,000. Instead, they decided to increase the amount given away every day
while still giving away the same total amount. If they want to increase the amount by
$100 each day, how much should they give away the first day?
1
a First term
n
a nth term
n
S sum of n terms
n number of terms
d common difference
a1
31 days
??
$124,000
$100/day
n 1 n
n
S a a
2
n 1
a a n 1 d
n 1 n
n
S a a
2
30. Sigma Notation ( )
Used to express a series or its sum in abbreviated form.
32.
j
4
1
j 2
2
1
2 2
3 2
2
4
18
7
4
a
2a
4
2
2 5
2 6
7
2
44
If the sequence is arithmetic (has a common difference) you can use the Sn formula
j
4
1
j 2
n 1 n
n
S a a
2
1
a First term
n
a nth term
n
S sum of n terms
n number of terms
d common difference
1+2=3
4
4+2=6
??
NA
33. Is the sequence arithmetic?
10 + 17 + 26 + 37
No, there is no common difference
Thus you cannot use the Sn formula.
= 90
= 2.71828
34. Rewrite using sigma notation: 3 + 6 + 9 + 12
Arithmetic, d= 3
n 1
a a n 1 d
n
a 3 n 1 3
n
a 3n
4
1
n
3n