This document discusses arithmetic sequences. It defines key terms like sequence, term, and common difference. It explains how to identify if a set of numbers forms an arithmetic sequence based on having a constant difference between terms. Methods are provided for finding the next term, the nth term, and representing the sequence as a linear function. Examples demonstrate how to apply these concepts to solve problems involving arithmetic sequences.
Algebra is used in many field in many different ways to solve equation problems, and in business algebra is also used or in our day to day life it is also used. ... Algebra is a way of keeping track of unknown values, which can be used in equations.
Algebra is used in many field in many different ways to solve equation problems, and in business algebra is also used or in our day to day life it is also used. ... Algebra is a way of keeping track of unknown values, which can be used in equations.
Step 1: Make sure the bottom numbers (the denominators) are the same
Step 2: Add the top numbers (the numerators), put that answer over the denominator
Step 3: Simplify the fraction (if possible)
Step 1. Make sure the bottom numbers (the denominators) are the same
Step 2. Subtract the top numbers (the numerators). Put the answer over the same denominator.
Step 3. Simplify the fraction (if needed).
This is a courseware on Algebraic Expression intended for high school teachers and students. It covers the concept and basic operations on algebraic expressions.
Contacts Details:
Mobile: +233 248870038
Email: ddeynu@aims.edu.gh, kwabla1991@gmail.com
Step 1: Make sure the bottom numbers (the denominators) are the same
Step 2: Add the top numbers (the numerators), put that answer over the denominator
Step 3: Simplify the fraction (if possible)
Step 1. Make sure the bottom numbers (the denominators) are the same
Step 2. Subtract the top numbers (the numerators). Put the answer over the same denominator.
Step 3. Simplify the fraction (if needed).
This is a courseware on Algebraic Expression intended for high school teachers and students. It covers the concept and basic operations on algebraic expressions.
Contacts Details:
Mobile: +233 248870038
Email: ddeynu@aims.edu.gh, kwabla1991@gmail.com
S&S Game is an Mathematics Game for Junior High School Students in year 8. It created in order to help teachers do an interactive learning, especially in sequences and series topic for grade 8. In this platform, it's only as a file review and uploaded in pdf format, so the macro designed in this game was unabled to show. If you mind to use the game, it's free to ask the creator for the pptm format of the game, so you can use the game perfectly.
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Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
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Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
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.
2. What’s on the Agenda
• Vocabulary
• Identifying arithmetic sequences
• Finding the next term of an arithmetic sequence
• Finding the nth term of an arithmetic sequence
• Arithmetic sequences and functions
3. Vocabulary
• A sequence is a set of numbers.
• The terms of a sequence are the specific order of the numbers in a set.
• An arithmetic sequence is a set of numbers where the difference between
two successive terms is constant.
• The difference between the two successive terms is known as the common
difference denoted d.
4. Identifying an Arithmetic Sequence
• Is the sequence of numbers an arithmetic sequence?
0, 2, 4, 6, 8, 10, …
• The number 0 is the first term, 2 is the second term, 4 is the third term etc.
• The difference between the 1st term and 2nd term, 2 – 0 = 2
• Because the difference between any two successive terms is 2, the sequence
has a common difference d = 2. So the sequence is an arithmetic sequence!
5. Identifying an Arithmetic Sequence
• Determine if the sequence is an arithmetic sequence.
−
1
2
, −
3
8
, −
1
8
, 0,
1
4
, …
• The difference between the first and second term is
1
8
. The difference
between the second and third term is
2
8
𝑜𝑟
1
4
. Since the two differences are
not equal this is not an arithmetic sequence.
6. Identifying an Arithmetic Sequence
• Determine if the sequence is an arithmetic sequence and explain why.
-20, -16, -12, -8, -4,…
7. Identifying an Arithmetic Sequence
• Determine if the sequence is an arithmetic sequence and explain why.
-20, -16, -12, -8, -4,…
• The difference between any two terms is 4.
• So the common difference d=4
• Since the difference between the terms in the sequence is constant this is an
arithmetic sequence.
8. Finding the Next Term
• If a sequence of numbers has a common difference d we can add d to the
previous term to get the next term.
• What are the 6th and 7th terms of the arithmetic sequence,
-13, -8, -3, 2, 7, …
• Step 1) Find the common difference d
• Step 2) Add d to the 5th term to get the 6th term
• Step 3) Add d to the 6th term to get the 7th term
9. Finding the Next Term
• What are the 6th and 7th terms of the arithmetic sequence,
-13, -8, -3, 2, 7, …
• Step 1) Find the common difference d
• d=5
• Step 2) Add d to the 5th term to get the 6th term
• The 5th term is 7 so 7+5=12
• 6th term is 12
• Step 3) Add d to the 6th term to get the 7th term
• 12+5=17
• 7th term is 17
10. Find the Next Term
• What are the next three terms in the arithmetic sequence?
-8.5, -9, -9.5, -10,…
11. Find the Next Term
• What are the next three terms in the arithmetic sequence?
-8.5, -9, -9.5, -10,…
• d=-0.5
• The 5th term is -10.5
• The 6th term is -11
• The 7th term is -11.5
12. Finding the nth Term
• So far we have figured out how to find next term from the sum of the previous
term and the common difference.
• We can actually find what any term is from the first term and the common
difference!
• To find the nth term of any arithmetic sequence with the first term 𝑎1and the
common difference d we can use the function
𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅
where n is a positive integer
13. Where does this equation come from!?
Term Symbol In Terms of 𝒂𝟏 and d In Terms of Successive
Terms and d
1st term 𝒂𝟏 𝒂𝟏 𝒂𝟏
2nd term 𝒂𝟐 𝒂𝟏 + 𝒅 𝒂𝟏 + 𝒅
3rd term 𝒂𝟑 𝒂𝟏 + 𝟐𝒅 (𝒂𝟏+𝒅) + 𝒅
=𝒂𝟐 + 𝒅
4th term 𝒂𝟒 𝒂𝟏 + 𝟑𝒅 (𝒂𝟐+𝒅) + 𝒅
=𝒂𝟑 + 𝒅
⋮ ⋮ ⋮ ⋮
nth term 𝒂𝒏 𝒂𝟏 + (𝒏 − 𝟏)𝒅 (𝒂(𝒏−𝟐)+𝒅) + 𝒅
=𝒂(𝒏−𝟏) + 𝒅
14. Finding the nth Term
• Write an equation for the nth term of the arithmetic sequence
-12, -8, -4, 0, …
• Step 1) Find the common difference
• 4
• Step 2) Write an equation
• 𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅
• 𝒂𝒏 = −𝟏𝟐 + 𝒏 − 𝟏 𝟒
• 𝒂𝒏 = −𝟏𝟐 + 𝟒𝒏 − 𝟒
• 𝒂𝒏 = 𝟒𝒏 − 𝟏𝟔
15. Finding the nth Term
• Find the 9th term of the sequence.
-12, -8, -4, 0, …
• Substitute 9 for n in the formula
• 𝒂𝒏 = 𝟒𝒏 − 𝟏𝟔
• 𝒂𝟗 = 𝟒(𝟗) − 𝟏𝟔
• 𝒂𝟗 = 𝟐𝟎
16. Finding the nth Term
• Graph the first five terms of the sequence.
-12, -8, -4, 0, …
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
0 1 2 3 4 5 6
n 4n-16 𝒂𝒏 (n, 𝒂𝒏)
1 4(1)-16 -12 (1,-12)
2 4(2)-16 -8 (2,-8)
3 4(3)-16 -4 (3,-4)
4 4(4)-16 0 (4,0)
5 4(5)-16 4 (5,4)
17. How to Graph on Desmos
(n, 𝒂𝒏)
(1,-12)
(2,-8)
(3,-4)
(4,0)
(5,4)
18. Finding the nth Term
• What term of the sequence is 32?
𝒂𝒏 = 𝟒𝒏 − 𝟏𝟔
32 = 𝟒(𝒏) − 𝟏𝟔
𝟑𝟐 + 𝟏𝟔 = 𝟒 𝒏 − 𝟏𝟔 + 𝟏𝟔
𝟒𝟖 = 𝟒 𝒏
𝟏𝟐 = 𝒏
• 32 is the 12th term of the sequence.
19. Finding the nth Term
• Consider the arithmetic sequence
3, -10, -23, -36,…
• 1) Write an equation for the nth term of the sequence
• 2) Find the 15th term in the sequence.
• 3) Graph the first 5 terms of the sequence.
• 4) Which term of the sequence is -114?
20. Finding the nth Term
• Consider the arithmetic sequence
3, -10, -23, -36,…
• 1) Write an equation for the nth term of the sequence
• d=-13
𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅
𝒂𝒏 = 𝟑 + 𝒏 − 𝟏 (−𝟏𝟑)
𝒂𝒏 = −𝟏𝟑𝒏 + 𝟏𝟔
21. Finding the nth Term
• Consider the arithmetic sequence
3, -10, -23, -36,…
• 2) Find the 15th term in the sequence.
𝒂𝒏 = −𝟏𝟑𝒏 + 𝟏𝟔
𝒂𝟏𝟓 = −𝟏𝟑(𝟏𝟓) + 𝟏𝟔
𝒂𝟏𝟓 = −𝟏𝟗𝟓 + 𝟏𝟔
𝒂𝟏𝟓 = −𝟏𝟕𝟗
• The 15th term of the sequence is -179.
22. Finding the nth Term
• Consider the arithmetic sequence
3, -10, -23, -36,…
• 3) Graph the first 5 terms of the sequence.
n −𝟏𝟑𝒏 + 𝟏𝟔 𝒂𝒏 (n, 𝒂𝒏)
1 -13(1)+16 3 (1,3)
2 -13(2)+16 -10 (2,-10)
3 -13(3)+16 -23 (3,-23)
4 -13(4)+16 -36 (4,-36)
5 -13(5)+16 -49 (5,-49)
23.
24. Finding the nth Term
• Consider the arithmetic sequence
3, -10, -23, -36,…
• 4) Which term of the sequence is -114?
−𝟏𝟏𝟒 = −𝟏𝟑𝒏 + 𝟏𝟔
−𝟏𝟏𝟒 − 𝟏𝟔 = −𝟏𝟑𝒏 + 𝟏𝟔 − 𝟏𝟔
−𝟏𝟑𝟎 = −𝟏𝟑𝒏
1𝟎 = 𝒏
• -114 is the 10th term in the sequence.
25. Arithmetic Sequences and Functions
• From the graph of an arithmetic sequence we see that arithmetic
sequences are linear functions.
• n is the x-value or independent variable
• 𝒂𝒏 is the y-value or dependent variable
• d the common difference is the slope.
• 𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅 as a linear functions is
f(n)= 𝒏 − 𝟏 𝒅 + 𝒂𝟏
26. Arithmetic Sequences as Functions
• Marisol is mailing invitations to her quinceañera. The arithmetic sequence
$0.42, $0.84, $1.26, $1.68,… represents the cost of postage.
• 1) Write a function to represent this sequence.
• 2) Graph the function
27. Arithmetic Sequences as Functions
• Marisol is mailing invitations to her quinceañera. The arithmetic sequence
$0.42, $0.84, $1.26, $1.68,… represents the cost of postage.
• 1) Write a function to represent this sequence.
• f(n)= 𝒏 − 𝟏 𝒅 + 𝒂𝟏
• f(n)= 𝒏 − 𝟏 𝟎. 𝟒𝟐 + 𝟎. 𝟒𝟐
• f(n)= 𝟎. 𝟒𝟐𝒏
28. Arithmetic Sequences as Functions
• Marisol is mailing invitations to her quinceañera. The arithmetic sequence
$0.42, $0.84, $1.26, $1.68,… represents the cost of postage.
• 2) Graph the function
n 0. 𝟒𝟐𝒏 f(n) (n, f(n))
1 0.42(1) 0.42 (1,0.42)
2 0.42(2) 0.84 (2,0.84)
3 0.42(3) 1.26 (3,1.26)
4 0.42(4) 1.68 (4,1.68)
5 0.42(5) 2.10 (5,1.90)
30. Review
• All arithmetic sequences have a ________ which is denoted ___.
• The common difference can be either a positive or negative number. (true or false)
• Arithmetic sequences are always direct variations. (true or false)
• You can find any term of an arithmetic sequence by ______ the common
difference to the previous term.
• The only things we need to determine the nth term of a sequence are the _____
term denoted 𝑎1 and the common difference.
31. THE BIG PICTURE
LINEAR FUNCTIONS
y=mx+b
ax+by=c
Direct Variation
y=kx and contains (0,0)
Arithmetic Sequences
𝑎𝑛 = 𝑛 − 1 𝑑 + 𝑎1
where n is positive