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Grade 7 Lesson about SEQUENCE AND SERIES
- 2. SEQUENCE AND SERIES A sequence is a set of numbers arranged in a specific order according to a rule, while a series is the sum of the terms in a sequence.
- 3. SEQUENCE AND SERIES Example: SEQUENCE In the arithmetic sequence 2,4,6,8,10,… the numbers are listed individually.
- 4. SEQUENCE AND SERIES Example: SERIES The series corresponding to the sequence 2,4,6,8,10 would be: 2+4+6+8+10=30
- 6. SEQUENCE ANDSERIES ARITHMETIC SEQUENCE An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference.
- 7. SEQUENCE ANDSERIES ARITHMETIC SEQUENCE n-th term: 𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑 Sum of the first n terms: 𝑆𝑛 = 𝑛 2 (2𝑎1 + 𝑛 − 1 𝑑)
- 8. SEQUENCE ANDSERIES ARITHMETIC SEQUENCE Example: Consider the sequence: 2, 5, 8, 11, 14 Common Difference (d) = 3 6th term (𝑎6) = 2+(6-1)3 = 17
- 9. SEQUENCE ANDSERIES ARITHMETIC SEQUENCE Example: Consider the sequence: 2, 5, 8, 11, 14 Sum of the first 5 terms: 𝑆5 = 5 2 (2(2) + 5 − 1 3) 𝑆5 = 5 2 4 + 12 = 40
- 10. SEQUENCE ANDSERIES ARITHMETIC SEQUENCE Try this one! Find the common difference, the 8th term, and the sum of the 8 terms. a. 25, 20, 15, 10, 5, …
- 11. SEQUENCE ANDSERIES ARITHMETIC SEQUENCE a. 25, 20, 15, 10, 5, ... Common Difference (d) = -5 𝑎8= 25+(8-1)(-5) 𝑎8= 25+(7)(-5) 𝑎8= 25+(-35) = -10
- 12. SEQUENCE ANDSERIES ARITHMETIC SEQUENCE a. 25, 20, 15, 10, 5, ... Common Difference (d) = -5 𝑆8 = 8 2 (2(25) + 8 − 1 (−5)) 𝑆8 = 8 2 50 + (−35) 𝑆8 = 8 2 15 = 60
- 14. SEQUENCE ANDSERIES GEOMETRIC SEQUENCE A geometric sequence is one in which the ratio between consecutive terms is constant. This ratio is called the common ratio.
- 15. SEQUENCE ANDSERIES GEOMETRIC SEQUENCE Common ratio: 𝑎2 𝑎1 n-th term: 𝑎𝑛 = 𝑎1(𝑟𝑛−1) Sum of the first n terms: 𝑆𝑛 = 𝑎1 1−𝑟𝑛 1−𝑟 (for r≠1)
- 16. SEQUENCE ANDSERIES GEOMETRIC SEQUENCE Example: Consider the sequence 3, 6, 12, 24, … Find the common ratio, 5th term, and sum of the 5 terms. Common ratio (r): 6/3 = 2 𝑎5 = 3 25−1 = 3 24 = 3 16 = 48
- 17. SEQUENCE ANDSERIES GEOMETRIC SEQUENCE Example: Consider the sequence 3, 6, 12, 24, … Find the common ratio, 5th term, and sum of the 5 terms. 𝑆5 = 3 1 − 25 1 − 2 𝑆5 = 3 −31 −1 = −63 −1 = 63
- 18. SEQUENCE ANDSERIES GEOMETRIC SEQUENCE Try this one! Find the common ratio, 6th term, and sum of the 6 terms. a. 81, 27, 9, 3, 1, …
- 19. SEQUENCE ANDSERIES GEOMETRIC SEQUENCE a. 81, 27, 9, 3, 1, … r = 27/81 = 1/3 𝑎6 = 81 1 3 6−1 = 81 1 3 5 = 81 1 243 = 1 3
- 20. SEQUENCE ANDSERIES GEOMETRIC SEQUENCE a. 81, 27, 9, 3, 1, … 𝑆6 = 81 1 − 1 3 6 1 − 1 3 𝑆6 = 81 1 − 1 729 2 3 = 81 728 729 2 3 𝑆6 = 121 1 3 𝑜𝑟 121.33
- 22. SEQUENCE ANDSERIES HARMONIC SEQUENCE A harmonic sequence is formed by taking the reciprocals of the terms of an arithmetic sequence. It is not an arithmetic or geometric sequence itself, but it has applications in physics and engineering.
- 23. SEQUENCE ANDSERIES HARMONIC SEQUENCE Example: Consider the arithmetic sequence 2, 4, 6, 8, … The corresponding harmonic sequence is: 1 2 , 1 4 , 1 6 , 1 8 , …
- 25. SEQUENCE ANDSERIES FIBONACCI SEQUENCE The Fibonacci sequence is a special sequence in which each term (after the first two) is the sum of the two preceding terms.
- 26. SEQUENCE ANDSERIES FIBONACCI SEQUENCE General Form: 0, 1, 1, 2, 3, 5, 8, 13, … n-th term: 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 where 𝐹0 = 0 and 𝐹1 = 1
- 27. SEQUENCE ANDSERIES FIBONACCI SEQUENCE 𝐹0 𝐹1 𝐹2 𝐹3 𝐹4 𝐹5 𝐹6 𝐹7 0 1 1 2 3 5 8 13 𝐹6 = 𝐹6−1 + 𝐹6−2 𝐹6 = 𝐹5 + 𝐹4 It means that the 6th term is the sum of the 5th and 4th terms.