Grade 10 Mathematics Arithmetic Series Lesson
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An arithmetic series is the sum of a sequence in which each term is computed from the previous one by adding (or subtracting) a constant. Arithmetic Sequence Formulas nth Term Formula an = a1 + (n – 1)d Sum of First n Terms Sn = n/2 (first term + last term)
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5.2 arithmetic sequences and sums t
The document provides examples and explanations of arithmetic sequences. It defines an arithmetic sequence as a sequence where the difference between consecutive terms is constant. The general formula for an arithmetic sequence is given as an = d(n-1) + a1, where d is the common difference and a1 is the first term. Several examples are worked through applying this formula to find the specific formula for given sequences, as well as the nth term and sum of terms. The exercises consist of identifying arithmetic sequences based on given terms and using the general formula to find characteristics like the first term, difference, specific formula, and sum of terms.
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The document discusses arithmetic sequences and series. Some key points:
1) An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This common difference (d) allows determining the nth term as an = dn + c, where c is the first term minus d.
2) The sum of the first n terms of a finite arithmetic sequence is given by S_n = (n/2)(a_1 + a_n), where a_1 is the first term and a_n is the nth term.
3) The sum of the first n terms of an infinite arithmetic sequence is called the nth partial sum. The partial sums can be represented using summation notation.
Arithmetic Sequence
This document defines arithmetic sequences and provides examples and explanations of key concepts such as the common difference, finding specific terms, finding sums of terms, and solving word problems involving arithmetic sequences. Key points covered include defining an arithmetic sequence as a sequence where each term is obtained by adding a constant to the preceding term, using the formula an = a1 + (n-1)d to find the nth term, and the formulas Sn = n(a1 + an)/2 and Sn = n(n-1)d/2 to find the sum of the first n terms. Examples of finding missing terms, common differences, and sums are worked through.
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The document provides information about arithmetic sequences and arithmetic series. It defines an arithmetic sequence as a sequence of numbers where each term after the first is obtained by adding the same constant to the previous term. It gives examples of arithmetic sequences and explains how to find the common difference, the nth term of a sequence using the general formula, and how to solve problems involving arithmetic sequences and series. The last paragraph tells a story about how Carl Friedrich Gauss was able to quickly calculate the sum of all numbers from 1 to 100 by recognizing it as an arithmetic series.
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The document defines an arithmetic sequence as a sequence where a constant difference can be added between each term to get the next term. It provides examples of finding the first term and common difference of arithmetic sequences. It explains that the common difference is found by subtracting any term from the one that follows it. Formulas are provided for finding specific terms in an arithmetic sequence given the first term, common difference, or other information. Examples problems are worked through applying these concepts.
Arithmetic progression
The document discusses arithmetic progressions (AP), which are sequences where the difference between successive terms is constant. It defines an AP using the notation an = a + (n-1)d, where a is the first term, d is the common difference, and n indexes the terms. It provides examples and explains how to find individual terms, determine if a sequence is an AP, and calculate the sum of terms in an AP using the formula Sn = 1/2n(2a + (n-1)d). Various problems are worked through to demonstrate applying the concepts and formulas for APs.
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The document discusses arithmetic and geometric sequences. It provides examples and explanations of key concepts such as common difference, common ratio, formulas for finding specific terms, and calculating series sums. Formulas are derived for finding the nth term in an arithmetic sequence as an = a1 + d(n-1) and the nth term in a geometric sequence as an = a1rn-1. Examples are worked through to demonstrate how to use the formulas and calculate sequence terms and series sums.
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1. The sum of the first 20 natural numbers is 210.
2. The sum of all odd natural numbers from 1 to 150 is 5625.
3. The sum of the terms of an AP with a = 6 and d = 3 up to the 10th term is 195.
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5.2 arithmetic sequences and sums t
The document provides examples and explanations of arithmetic sequences. It defines an arithmetic sequence as a sequence where the difference between consecutive terms is constant. The general formula for an arithmetic sequence is given as an = d(n-1) + a1, where d is the common difference and a1 is the first term. Several examples are worked through applying this formula to find the specific formula for given sequences, as well as the nth term and sum of terms. The exercises consist of identifying arithmetic sequences based on given terms and using the general formula to find characteristics like the first term, difference, specific formula, and sum of terms.
Arithmetic Sequence and Series
The document discusses arithmetic sequences and series. Some key points:
1) An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This common difference (d) allows determining the nth term as an = dn + c, where c is the first term minus d.
2) The sum of the first n terms of a finite arithmetic sequence is given by S_n = (n/2)(a_1 + a_n), where a_1 is the first term and a_n is the nth term.
3) The sum of the first n terms of an infinite arithmetic sequence is called the nth partial sum. The partial sums can be represented using summation notation.
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This document defines arithmetic sequences and provides examples and explanations of key concepts such as the common difference, finding specific terms, finding sums of terms, and solving word problems involving arithmetic sequences. Key points covered include defining an arithmetic sequence as a sequence where each term is obtained by adding a constant to the preceding term, using the formula an = a1 + (n-1)d to find the nth term, and the formulas Sn = n(a1 + an)/2 and Sn = n(n-1)d/2 to find the sum of the first n terms. Examples of finding missing terms, common differences, and sums are worked through.
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The document defines an arithmetic sequence as a sequence where a constant difference can be added between each term to get the next term. It provides examples of finding the first term and common difference of arithmetic sequences. It explains that the common difference is found by subtracting any term from the one that follows it. Formulas are provided for finding specific terms in an arithmetic sequence given the first term, common difference, or other information. Examples problems are worked through applying these concepts.
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The document discusses arithmetic and geometric sequences. It provides examples and explanations of key concepts such as common difference, common ratio, formulas for finding specific terms, and calculating series sums. Formulas are derived for finding the nth term in an arithmetic sequence as an = a1 + d(n-1) and the nth term in a geometric sequence as an = a1rn-1. Examples are worked through to demonstrate how to use the formulas and calculate sequence terms and series sums.
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1. The sum of the first 20 natural numbers is 210.
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If you are looking for math video tutorials (with voice recording), you may download it on our YouTube Channel. Don't forget to SUBSCRIBE for you to get updated on our upcoming videos.
https://tinyurl.com/y9muob6q
Also, please do visit our page, LIKE and FOLLOW us on Facebook!
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Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
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Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Grade 10 Mathematics Arithmetic Series Lesson
- 2. There was once a schoolboy who was asked by his teacher to give the sum of the first 100 counting numbers. 1 + 2 + 3 + … + 98 + 99 + 100
- 3. 1 + 2 + 3 +…+ 50 + 51 +… + 98 + 99 + 100 101 101 101 101 101 _x 50 pairings_ 5,050
- 4. JOHANN CARL FRIEDRICH GAUSS
- 5. ARITHMETIC SERIES It is the sum of an arithmetic sequence
- 6. ARITHMETIC SERIES • A series such as (3 + 7 + 11 + 15 + ··· + 99 or 10 + 20 + 30 + ··· + 1000) which has a constant difference between terms. first term is a1 common difference is d number of terms is n last term is an sum of an arithmetic series is Sn
- 7. ARITHMETIC SERIES The sum of an arithmetic series is found by multiplying the number of terms times the average of the first and last terms.
- 8. ARITHMETIC SERIES Sn = 𝒏 𝟐 (a1 + an) Formula ( FINITE SEQUENCE) ~ when an is given
- 9. ARITHMETIC SERIES Formula (INFINITE SEQUENCE) Sn = 𝐧 𝟐 [2a1 + (n - 1) d]
- 10. ARITHMETIC SERIES 3 + 8 + 13 + … + 73 • a1 = 3 • an = 73 • d = 5 • n = ? • Sn = ? Example 1
- 11. ARITHMETIC SERIES Sn = 𝒏 𝟐 (a1 + an) S15 = 𝟏𝟓 𝟐 (3 + 73) S15 = 𝟏𝟓 𝟐 (76) S15 = 𝟏𝟓 (38) S15 = 57𝟎 Example 1
- 12. ARITHMETIC SERIES 3 + 7 + 11 + 15 + ··· + 99
- 13. ARITHMETIC SERIES 𝟏 𝟑 + 𝟐 𝟑 + 1 +··· + 𝟏𝟔 𝟑
- 14. ARITHMETIC SERIES Find the sum of the sequence -8, -5, -2, ..., 7
- 15. •1. integers from 1 to 50 •2. odd integers from 1 to 100 •3. even integers between 1 and 101 •9. 1 + 5 + 9 + … + 49 + 53 •10. 𝟏 𝟐 + 𝟑 𝟐 + 𝟓 𝟐 +… 𝟏𝟕 𝟐 + 𝟏𝟗 𝟐
- 16. 1. INTEGERS FROM 1 TO 50 Given: a1=1 an=50 d=1 n= 50 Sn =1275 n=? Sn=?
- 17. 2. ODD INTEGERS FROM 1 TO 100 Given: a1=1 an=99 d=2 n= 50 Sn =2500 n=? Sn=?
- 18. 3. EVEN INTEGERS BETWEEN 1 AND 101 Given: a1=2 an=100 d=2 n= 50 Sn =2550 n=? Sn=?
- 19. 9.) 1 + 5 + 9 + … + 49 + 53 Given: a1=1 an=53 d=4 n= 14 Sn =378 n=? Sn=?
- 21. ARITHMETIC SERIES Formula (INFINITE SEQUENCE) Sn = 𝐧 𝟐 [2a1 + (n - 1) d]
- 22. ARITHMETIC SERIES What is the sum of the first 10 terms in the series: 3, 7, 11, … •a1 = 3 •d = 4 • n = 10 •Sn = ? Example
- 23. ARITHMETIC SERIES Sn = 𝐧 𝟐 [2a1 + (n - 1) d] S10 = 𝟏𝟎 𝟐 [2(3) + (10 - 1) 4] S10 = 𝟏𝟎 𝟐 [6 + (9) 4] S10 = 𝟏𝟎 𝟐 [6 + 36] S10 = 𝟏𝟎 𝟐 [42] Solution 1 S10 = 𝟒𝟐𝟎 𝟐 S10 = 𝟐𝟏𝟎
- 24. ARITHMETIC SERIES What is the sum of the first 10 terms in the series: 3, 7, 11, … •a1 = 3 •d = 4 • n = 10 • an = ? • Sn = ? Solution 2
- 25. ARITHMETIC SERIES an = a1 + (n – 1) d a10 = 3 + (10 – 1) 4 a10 = 3 + (9) 4 a10 = 3 + 36 a10 = 39 Solution 2
- 26. ARITHMETIC SERIES Sn = 𝒏 𝟐 (a1 + an) S10 = 𝟏𝟎 𝟐 (3 + 39) S10 = 5 (42) S10 = 𝟐𝟏𝟎 Solution 2
- 27. ARITHMETIC SERIES Find the sum of the first 12 positive even integers.
- 28. ARITHMETIC SERIES Find the sum of the first 10 negative integers
- 29. ARITHMETIC SERIES Find the sum of the first 20 terms of the sequence 4, 6, 8, 10, ...
- 30. ARITHMETIC SERIES Find the sum of the first nine terms of the arithmetic sequence with 𝒂𝟏 = −𝟐 and 𝒅 = 𝟓.
- 31. ARITHMETIC SERIES Find the sum of the first five terms of the arithmetic sequence with 𝒂𝟏 = 𝟏 𝟒 and 𝒅 = −𝟏.