Mathematics 10
Arithmetic Means and nth
Term of an Arithmetic
Sequence
Quarter 1 Module 3Lesson 1
INDEX
2. Greetings
1. Opening Prayer
3. Checking of Attendance
6. Review
5. Objectives
4. Motivation
7. Lesson Proper (Lesson 1)
9. Activity
8. Assessment
10. Review
11. Lesson Proper (Lesson 2)
13. Assignment
12. Assessment
14. References
Let's bow our head and feel the presence of our Lord...
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Good Day Class!
Teacher Joanna Marie F. Garcia
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WE MUST FOLLOW THE ORDER...
TO SUCCEED
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2. use the formula to find the nth term or unknown term of an arithmetic sequence,
1. write a formula for the nth term of an arithmetic sequence,
Objectives
4. determine arithmetic means of a sequence.
3. define arithmetic means, and
Back
review
Back
What about if the problem is to find the 100th term or the 250th term? Can you find the terms? Using the process that is illustrated above will take much of your time and effort. There is a short cut in doing this and that is one of the focus of this module.
Before we find other higher terms of a sequence, let us first find lower terms. In the arithmetic sequence: 3, 8, 13, 18,…; what is the 15th term?
Solution:
a. By adding the common difference to each of the preceding terms, we get the following values.
b. Thus, the 15th term is 73.
However, using this procedure to get any higher nth term would be tedious. Thus, a formula is necessary to find any nth term.
example:
Let us investigate on how to determine the nth term of a sequence. In the table:
a1 = 3 = 3
a2 = 3 + 5 = 8
a3 = 3 + 5 + 5 = 13
a4 = 3 + 5 + 5 + 5 = 18
. .
. .
. .
a13 = 3 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 63
a14 = 3 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 68
a15 = 3 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 73
These terms can be written in the following manner as a short cut.
a1 = 3 = 3
a2 = 3 + 5 (1) = 8
a3 = 3 + 5 (2) = 13
a4 = 3 + 5 (3) = 18
. .
. .
. .
a13 = 3 + 5 (12) = 63
a14 = 3 + 5 (13) = 68
a15 = 3 + 5 (14) = 73
Thus, if we find for the 16th term of the arithmetic sequence, then a16 = 3
+ 5 (15) = 78. We can conclude that using the pattern observed the nth term of the sequence is an = a1 + d (n-1), where an is the term that corresponds to nth position, a1 is the first term, and d is the common difference.
let's try!
Let us apply this formula in solving the following: A. Find the 21st term of the arithmetic sequence: 6, 9, 12, 15,…
Info
let's try!
B. In the arithmetic sequence: 7, 10, 13, 16, . . .; find n if an = 304.
Info
Back
ASSESSMENT:
A. Find the specified nth term of each arithmetic sequence. 1. 2, 5, 8, …; 9th term
2. 3, 5 7, …; 20th term
3. 1, 1/2, 0, …; 16th term
4. 5, 11, 17, …; 9th term 5. 26, 22, 18, …; 40th term
6. 103rd term of the arithmetic sequence if a1 = -5 and d = -4
7. 19th term of the arithmetic sequence if a1 = 25 and d = -2
8. 25th term of the arithmetic sequence if a = 1/2 and d = — 3/8.
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ACTIVITY:
A. Give what is asked: 1. The 10th term of the arithmetic sequence if a1 = -15 and d = 6 2. The 39th term of the arithmetic sequence if a = 40 and d = 1/2 B. Find the specified term of each arithmetic sequence. 1. 1.4, 4.5, 7.6, …; The 41st term 2. 9, 18, 27,…; the 23rd term 3. 14, 6, -2,…; 27th term 4. 3, 3.25, 3.5,…; 16th term 5. 1, 4, 7,…; 28th term
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Mathematics 10
COMPUTING ARITHMETIC MEANS
Quarter 1 Module 3Lesson 2
REVIEW
In the previous lesson, you learned how to determine the nth term of an arithmetic sequence.For example: In the sequence: 10, 15, 20, 25,…; what term has a value of 385?
Solution:
a. Using the formula, an = a1 + d(n – 1):
385 = 10 + 5 ( n – 1 )
385 = 10 + 5n -5
385 = 5n + 5
5n = 385 – 5
5n = 380
n = 76
b. Thus, 385 is the 76th term of the given sequence.
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The first and last terms of a finite arithmetic sequences are called arithmetic extremes, and the terms in between are called arithmetic means. In the sequence 4, 8, 12, 16, 20, 24; the terms 4 and 24 are the arithmetic extremes, while 8, 12, 16, and 20 are the arithmetic means. Also, 8 is the arithmetic mean of the arithmetic extremes, 4 and 12.
The arithmetic mean between two numbers is sometimes called the average of two numbers. If more than one arithmetic means will be inserted between two arithmetic extremes, the formula for d, which is d = an—ak/n-k, can be used.
LET'S TRY!
A. What is the arithmetic mean between 10 and 24?
LET'S TRY!
B. Insert three arithmetic means between 8 and 16.
Back
assessment
Choose the letter of your answer from the given options. Write your answer on a separate paper.
1. Which term of the arithmetic sequence 5, 9, 13, 17, … is 409?
a. 99th term b. 100th term c. 111th term d. 102th term
2. Find the nth term of the arithmetic sequence given the following conditions: a1=5 d= 5 n=25 a. 25th term=115 b. 25th term=125 c. 25th term=120 d. 25th term=130
3. Which term of the arithmetic sequence 5, 9, 13, 17,….. is 401?
a. 99th term b. 100th term c. 111th term d. 112th term
assessment
4. If three arithmetic means are inserted between -15 and 9, find the first of these arithmetic means.
a. 3 b. -3 c. -6 d. -9 5. Find the 20th term of the arithmetic sequence 5, 9, 13, 17, 21,… a. 81 b. 80 c. 82 d. 87
6. If three arithmetic means are inserted between 11 and 39, find the second arithmetic mean.
a. 18 b. 25 c. 32 d. 46
7. Which term of the arithmetic sequence 4, 1, -2, -5, … is -29?
a. 9th term b. 10th term c. 11th term d. 12th term
assessment
8. What is the arithmetic mean between 15 and 40?
a. 28.5 b. 29 c. 26 d. 27.5
9. What is the 8th term of the following arithmetic sequence: -5, -1, 3, 7, 11,…?
a. 23 b. 19 c. 27 d. 22 10. If a1 = -3 and a5 = 5. Find a10?
a. 14 b. 15 c. 16 d. 17
assessment
8. What is the arithmetic mean between 15 and 40?
a. 28.5 b. 29 c. 26 d. 27.5
9. What is the 8th term of the following arithmetic sequence: -5, -1, 3, 7, 11,…?
a. 23 b. 19 c. 27 d. 22 10. If a1 = -3 and a5 = 5. Find a10?
a. 14 b. 15 c. 16 d. 17
Back
assignment
Solve the following word problems correctly. Give what is asked.
1. You have accepted a job with a salary of P27,000.00 a month during the first year. At the end of each year, you receive a P1,500.00 raise. What is your monthly salary during the first six years?
2. An object is dropped from a plane and falls 32 feet during the first second. For each succeeding second, it falls 40 feet more than the distance covered in the preceding second. How far has it fallen after 11 seconds?
Back
Thankseveryone!
references
Callanta, Melvin M., et al.2015. Mathematics Learner’s Module.Pasig City.
Nivera, Gladys C. and Lapinid, Minie Rose C.2015,Grade 10 Mathematics: Patterns and Practicalities. Makati City, Don Bosco Press.
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ARITHMETIC MEAN AND NTH TERM OF AN ARITHMETIC SEQUENCE
Joanna Garcia
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Transcript
Mathematics 10
Arithmetic Means and nth Term of an Arithmetic Sequence
Quarter 1 Module 3Lesson 1
INDEX
2. Greetings
1. Opening Prayer
3. Checking of Attendance
6. Review
5. Objectives
4. Motivation
7. Lesson Proper (Lesson 1)
9. Activity
8. Assessment
10. Review
11. Lesson Proper (Lesson 2)
13. Assignment
12. Assessment
14. References
Let's bow our head and feel the presence of our Lord...
Back
Good Day Class!
Teacher Joanna Marie F. Garcia
Back
WE MUST FOLLOW THE ORDER...
TO SUCCEED
Back
2. use the formula to find the nth term or unknown term of an arithmetic sequence,
1. write a formula for the nth term of an arithmetic sequence,
Objectives
4. determine arithmetic means of a sequence.
3. define arithmetic means, and
Back
review
Back
What about if the problem is to find the 100th term or the 250th term? Can you find the terms? Using the process that is illustrated above will take much of your time and effort. There is a short cut in doing this and that is one of the focus of this module.
Before we find other higher terms of a sequence, let us first find lower terms. In the arithmetic sequence: 3, 8, 13, 18,…; what is the 15th term? Solution: a. By adding the common difference to each of the preceding terms, we get the following values.
b. Thus, the 15th term is 73. However, using this procedure to get any higher nth term would be tedious. Thus, a formula is necessary to find any nth term.
example:
Let us investigate on how to determine the nth term of a sequence. In the table: a1 = 3 = 3 a2 = 3 + 5 = 8 a3 = 3 + 5 + 5 = 13 a4 = 3 + 5 + 5 + 5 = 18 . . . . . . a13 = 3 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 63 a14 = 3 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 68 a15 = 3 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 73
These terms can be written in the following manner as a short cut. a1 = 3 = 3 a2 = 3 + 5 (1) = 8 a3 = 3 + 5 (2) = 13 a4 = 3 + 5 (3) = 18 . . . . . . a13 = 3 + 5 (12) = 63 a14 = 3 + 5 (13) = 68 a15 = 3 + 5 (14) = 73 Thus, if we find for the 16th term of the arithmetic sequence, then a16 = 3 + 5 (15) = 78. We can conclude that using the pattern observed the nth term of the sequence is an = a1 + d (n-1), where an is the term that corresponds to nth position, a1 is the first term, and d is the common difference.
let's try!
Let us apply this formula in solving the following: A. Find the 21st term of the arithmetic sequence: 6, 9, 12, 15,…
Info
let's try!
B. In the arithmetic sequence: 7, 10, 13, 16, . . .; find n if an = 304.
Info
Back
ASSESSMENT:
A. Find the specified nth term of each arithmetic sequence. 1. 2, 5, 8, …; 9th term 2. 3, 5 7, …; 20th term 3. 1, 1/2, 0, …; 16th term 4. 5, 11, 17, …; 9th term 5. 26, 22, 18, …; 40th term 6. 103rd term of the arithmetic sequence if a1 = -5 and d = -4 7. 19th term of the arithmetic sequence if a1 = 25 and d = -2 8. 25th term of the arithmetic sequence if a = 1/2 and d = — 3/8.
Back
ACTIVITY:
A. Give what is asked: 1. The 10th term of the arithmetic sequence if a1 = -15 and d = 6 2. The 39th term of the arithmetic sequence if a = 40 and d = 1/2 B. Find the specified term of each arithmetic sequence. 1. 1.4, 4.5, 7.6, …; The 41st term 2. 9, 18, 27,…; the 23rd term 3. 14, 6, -2,…; 27th term 4. 3, 3.25, 3.5,…; 16th term 5. 1, 4, 7,…; 28th term
Back
Mathematics 10
COMPUTING ARITHMETIC MEANS
Quarter 1 Module 3Lesson 2
REVIEW
In the previous lesson, you learned how to determine the nth term of an arithmetic sequence.For example: In the sequence: 10, 15, 20, 25,…; what term has a value of 385? Solution: a. Using the formula, an = a1 + d(n – 1): 385 = 10 + 5 ( n – 1 ) 385 = 10 + 5n -5 385 = 5n + 5 5n = 385 – 5 5n = 380 n = 76 b. Thus, 385 is the 76th term of the given sequence.
Back
The first and last terms of a finite arithmetic sequences are called arithmetic extremes, and the terms in between are called arithmetic means. In the sequence 4, 8, 12, 16, 20, 24; the terms 4 and 24 are the arithmetic extremes, while 8, 12, 16, and 20 are the arithmetic means. Also, 8 is the arithmetic mean of the arithmetic extremes, 4 and 12.
The arithmetic mean between two numbers is sometimes called the average of two numbers. If more than one arithmetic means will be inserted between two arithmetic extremes, the formula for d, which is d = an—ak/n-k, can be used.
LET'S TRY!
A. What is the arithmetic mean between 10 and 24?
LET'S TRY!
B. Insert three arithmetic means between 8 and 16.
Back
assessment
Choose the letter of your answer from the given options. Write your answer on a separate paper. 1. Which term of the arithmetic sequence 5, 9, 13, 17, … is 409? a. 99th term b. 100th term c. 111th term d. 102th term 2. Find the nth term of the arithmetic sequence given the following conditions: a1=5 d= 5 n=25 a. 25th term=115 b. 25th term=125 c. 25th term=120 d. 25th term=130 3. Which term of the arithmetic sequence 5, 9, 13, 17,….. is 401? a. 99th term b. 100th term c. 111th term d. 112th term
assessment
4. If three arithmetic means are inserted between -15 and 9, find the first of these arithmetic means. a. 3 b. -3 c. -6 d. -9 5. Find the 20th term of the arithmetic sequence 5, 9, 13, 17, 21,… a. 81 b. 80 c. 82 d. 87 6. If three arithmetic means are inserted between 11 and 39, find the second arithmetic mean. a. 18 b. 25 c. 32 d. 46 7. Which term of the arithmetic sequence 4, 1, -2, -5, … is -29? a. 9th term b. 10th term c. 11th term d. 12th term
assessment
8. What is the arithmetic mean between 15 and 40? a. 28.5 b. 29 c. 26 d. 27.5 9. What is the 8th term of the following arithmetic sequence: -5, -1, 3, 7, 11,…? a. 23 b. 19 c. 27 d. 22 10. If a1 = -3 and a5 = 5. Find a10? a. 14 b. 15 c. 16 d. 17
assessment
8. What is the arithmetic mean between 15 and 40? a. 28.5 b. 29 c. 26 d. 27.5 9. What is the 8th term of the following arithmetic sequence: -5, -1, 3, 7, 11,…? a. 23 b. 19 c. 27 d. 22 10. If a1 = -3 and a5 = 5. Find a10? a. 14 b. 15 c. 16 d. 17
Back
assignment
Solve the following word problems correctly. Give what is asked. 1. You have accepted a job with a salary of P27,000.00 a month during the first year. At the end of each year, you receive a P1,500.00 raise. What is your monthly salary during the first six years? 2. An object is dropped from a plane and falls 32 feet during the first second. For each succeeding second, it falls 40 feet more than the distance covered in the preceding second. How far has it fallen after 11 seconds?
Back
Thankseveryone!
references
Callanta, Melvin M., et al.2015. Mathematics Learner’s Module.Pasig City. Nivera, Gladys C. and Lapinid, Minie Rose C.2015,Grade 10 Mathematics: Patterns and Practicalities. Makati City, Don Bosco Press.
Back