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### 91 sequences

1. 1. Sequences
2. 2. SequencesA sequence is an ordered list of infinitely manynumbers that may or may not have a pattern.
3. 3. SequencesA sequence is an ordered list of infinitely manynumbers that may or may not have a pattern.Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.
4. 4. SequencesA sequence is an ordered list of infinitely manynumbers that may or may not have a pattern.Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.
5. 5. SequencesA sequence is an ordered list of infinitely manynumbers that may or may not have a pattern.Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obviouspattern.
6. 6. SequencesA sequence is an ordered list of infinitely manynumbers that may or may not have a pattern.Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obviouspattern.Definition: A sequence is the list of outputs f(n) of afunction f where n = 1, 2, 3, 4, ….. We write f(n) as fn.
7. 7. SequencesA sequence is an ordered list of infinitely manynumbers that may or may not have a pattern.Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obviouspattern.Definition: A sequence is the list of outputs f(n) of afunction f where n = 1, 2, 3, 4, ….. We write f(n) as fn.A sequence may be listed as f1, f2 , f3 , …
8. 8. SequencesA sequence is an ordered list of infinitely manynumbers that may or may not have a pattern.Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obviouspattern.Definition: A sequence is the list of outputs f(n) of afunction f where n = 1, 2, 3, 4, ….. We write f(n) as fn.A sequence may be listed as f1, f2 , f3 , …f100 = 100th number on the list,
9. 9. SequencesA sequence is an ordered list of infinitely manynumbers that may or may not have a pattern.Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obviouspattern.Definition: A sequence is the list of outputs f(n) of afunction f where n = 1, 2, 3, 4, ….. We write f(n) as fn.A sequence may be listed as f1, f2 , f3 , …f100 = 100th number on the list,fn = the n’th number on the list,
10. 10. SequencesA sequence is an ordered list of infinitely manynumbers that may or may not have a pattern.Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obviouspattern.Definition: A sequence is the list of outputs f(n) of afunction f where n = 1, 2, 3, 4, ….. We write f(n) as fn.A sequence may be listed as f1, f2 , f3 , …f100 = 100th number on the list,fn = the n’th number on the list,fn-1 = the (n – 1)’th number on the list or the numberbefore fn.
11. 11. SequencesExample B:a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25,
12. 12. SequencesExample B:a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.
13. 13. SequencesExample B:a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10,
14. 14. SequencesExample B:a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*n
15. 15. SequencesExample B:a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*nc. For the sequence of odd numbers 1, 3, 5, … a general formula is fn= 2n – 1.
16. 16. SequencesExample B:a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*nc. For the sequence of odd numbers 1, 3, 5, … a general formula is fn= 2n – 1.
17. 17. SequencesExample B:a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*nc. For the sequence of odd numbers 1, 3, 5, … a general formula is fn= 2n – 1.d. For the sequence of odd numbers with alternating signs -1, 3, -5, 7, -9, …fn = (-1)n(2n – 1)
18. 18. SequencesExample B:a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*nc. For the sequence of odd numbers 1, 3, 5, … a general formula is fn= 2n – 1.d. For the sequence of odd numbers with alternating signs -1, 3, -5, 7, -9, …fn = (-1)n(2n – 1)A sequence whose signs alternate is called analternating sequence as in part d.
19. 19. Summation Notation
20. 20. Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”.
21. 21. Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,
22. 22. Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s.
23. 23. Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s. (Of course the xsand xys have to be given in the context.)
24. 24. Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s. (Of course the xsand xys have to be given in the context.)Given a list, lets say, of 100 numbers, f1, f2, f3,.., f100,
25. 25. Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s. (Of course the xsand xys have to be given in the context.)Given a list, lets say, of 100 numbers, f1, f2, f3,.., f100,their sum f1 + f2 + f3 ... + f99 + f100 may be written inthe - notation as: 100 k=1 fk
26. 26. Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s. (Of course the xsand xys have to be given in the context.)Given a list, lets say, of 100 numbers, f1, f2, f3,.., f100,their sum f1 + f2 + f3 ... + f99 + f100 may be written inthe - notation as: 100 k=1 fk A variable which is called the “index” variable, in this case k.
27. 27. Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s. (Of course the xsand xys have to be given in the context.)Given a list, lets say, of 100 numbers, f1, f2, f3,.., f100,their sum f1 + f2 + f3 ... + f99 + f100 may be written inthe - notation as: 100 k=1 fk A variable which is called the “index” variable, in this case k. k begins with the bottom number and counts up (runs) to the top number.
28. 28. Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s. (Of course the xsand xys have to be given in the context.)Given a list, lets say, of 100 numbers, f1, f2, f3,.., f100,their sum f1 + f2 + f3 ... + f99 + f100 may be written inthe - notation as: 100 k=1 fk A variable which is called the The beginning number “index” variable, in this case k. k begins with the bottom number and counts up (runs) to the top number.
29. 29. Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s. (Of course the xsand xys have to be given in the context.)Given a list, lets say, of 100 numbers, f1, f2, f3,.., f100,their sum f1 + f2 + f3 ... + f99 + f100 may be written inthe - notation as: The ending number 100 k=1 fk A variable which is called the The beginning number “index” variable, in this case k. k begins with the bottom number and counts up (runs) to the top number.
30. 30. Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s. (Of course the xsand xys have to be given in the context.)Given a list, lets say, of 100 numbers, f1, f2, f3,.., f100,their sum f1 + f2 + f3 ... + f99 + f100 may be written inthe - notation as: The ending number 100 k=1 fk = f1 f2 f3 … f99 f100 A variable which is called the The beginning number “index” variable, in this case k. k begins with the bottom number and counts up (runs) to the top number.
31. 31. Summation NotationIn mathematics, the Greek letter “ ” (sigma) means“to add”. Hence, “ x” means to add the x’s,“ (x*y)” means to add the x*y’s. (Of course the xsand xys have to be given in the context.)Given a list, lets say, of 100 numbers, f1, f2, f3,.., f100,their sum f1 + f2 + f3 ... + f99 + f100 may be written inthe - notation as: The ending number 100 k=1 fk = f1+ f2+ f3+ … + f99+ f100 A variable which is called the The beginning number “index” variable, in this case k. k begins with the bottom number and counts up (runs) to the top number.
32. 32. Summation Notation 8k=4 fk = 5 ai =i=2 9 xjyj = j=6n+3 aj =j=n
33. 33. Summation NotationExample C: 8k=4 fk = f 4+ f5+ f6+ f7 + f 8 5 ai =i=2 9 xjyj = j=6n+3 aj =j=n
34. 34. Summation NotationExample C: 8k=4 fk = f 4+ f5+ f6+ f7 + f 8 5 a i = a 2+ a 3+ a 4+ a 5i=2 9 xjyj = j=6n+3 aj =j=n
35. 35. Summation NotationExample C: 8k=4 fk = f 4+ f5+ f6+ f7 + f 8 5 a i = a 2+ a 3+ a 4+ a 5i=2 9 x j y j = x 6y 6+ x 7y 7 + x 8y 8+ x 9y 9 j=6n+3 aj =j=n
36. 36. Summation NotationExample C: 8k=4 fk = f 4+ f5+ f6+ f7 + f 8 5 a i = a 2+ a 3+ a 4+ a 5i=2 9 x j y j = x 6y 6+ x 7y 7 + x 8y 8+ x 9y 9 j=6n+3 aj = an+ an+1+ an+2+ an+3j=n
37. 37. Summation NotationExample C: 8k=4 fk = f 4+ f5+ f6+ f7 + f 8 5 a i = a 2+ a 3+ a 4+ a 5i=2 9 x j y j = x 6y 6+ x 7y 7 + x 8y 8+ x 9y 9 j=6n+3 aj = an+ an+1+ an+2+ an+3j=nSummation notation are used to express formulas inmathematics.
38. 38. Summation NotationExample C: 8k=4 fk = f 4+ f5+ f6+ f7 + f 8 5 a i = a 2+ a 3+ a 4+ a 5i=2 9 x j y j = x 6y 6+ x 7y 7 + x 8y 8+ x 9y 9 j=6n+3 aj = an+ an+1+ an+2+ an+3j=nSummation notation are used to express formulas inmathematics. An example is the formula foraveraging.
39. 39. Summation NotationExample C: 8k=4 fk = f 4+ f5+ f6+ f7 + f 8 5 a i = a 2+ a 3+ a 4+ a 5i=2 9 x j y j = x 6y 6+ x 7y 7 + x 8y 8+ x 9y 9 j=6n+3 aj = an+ an+1+ an+2+ an+3j=nSummation notation are used to express formulas inmathematics. An example is the formula foraveraging. Given n numbers, f1, f2, f3,.., fn, theiraverage (mean), written as f,is (f1 + f2 + f3 ... + fn-1 + fn)/n.
40. 40. Summation NotationExample C: 8k=4 fk = f 4+ f5+ f6+ f7 + f 8 5 a i = a 2+ a 3+ a 4+ a 5i=2 9 x j y j = x 6y 6+ x 7y 7 + x 8y 8+ x 9y 9 j=6n+3 aj = an+ an+1+ an+2+ an+3j=nSummation notation are used to express formulas inmathematics. An example is the formula foraveraging. Given n numbers, f1, f2, f3,.., fn, their naverage (mean), written as f, f k=1 kis (f1 + f2 + f3 ... + fn-1 + fn)/n. In notation, f = n
41. 41. Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.
42. 42. Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) k=5
43. 43. Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = k=5 k=5 k=6 k=7 k=8
44. 44. Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) k=5 k=5 k=6 k=7 k=8
45. 45. Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) k=5 k=5 k=6 k=7 k=8 = 24 + 35 + 48 + 63
46. 46. Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) k=5 k=5 k=6 k=7 k=8 = 24 + 35 + 48 + 63 = 170
47. 47. Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) k=5 k=5 k=6 k=7 k=8 = 24 + 35 + 48 + 63 5 = 170b. (-1)k(3k + 2) k=3
48. 48. Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) k=5 k=5 k=6 k=7 k=8 = 24 + 35 + 48 + 63 5 = 170b. (-1)k(3k + 2) k=3 =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2)
49. 49. Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) k=5 k=5 k=6 k=7 k=8 = 24 + 35 + 48 + 63 5 = 170b. (-1)k(3k + 2) k=3 =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2) = -11 + 14 – 17
50. 50. Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) k=5 k=5 k=6 k=7 k=8 = 24 + 35 + 48 + 63 5 = 170b. (-1)k(3k + 2) k=3 =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2) = -11 + 14 – 17 = -14
51. 51. Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) k=5 k=5 k=6 k=7 k=8 = 24 + 35 + 48 + 63 5 = 170b. (-1)k(3k + 2) k=3 =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2) = -11 + 14 – 17 = -14In part b, the multiple (-1)k change the sums to analternating sum, t
52. 52. Summation NotationThe index variable is also used as the variable thatgenerates the numbers to be summed.Example D: 8a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) k=5 k=5 k=6 k=7 k=8 = 24 + 35 + 48 + 63 5 = 170b. (-1)k(3k + 2) k=3 =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2) = -11 + 14 – 17 = -14In part b, the multiple (-1)k change the sums to analternating sum, that is, a sum where the termsalternate between positive and negative numbers.