Chapter 6 Sequences

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Chapter 6Sequences
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Sequences, Strings, Summations

• ?????(sequence) ??????? ?????????????????????????
  ?????????????????????????????
• ?????????????????????????????????(index)
  ?????????????
• ???????????????????????????????????????
• ?????????(string) ????????????????????????????????
  ?????? ???? science ??? sceenic
  ???????????????????????? ?????????????????????????
  ??????? ????????????????????????????
• ??????(series) ??????? ?????????????????
  ????????????????????????????

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Sequences

• ?????(Sequences) ?????????????????????????????????
  ?
• ??????????????????????????????? N ???????? S
  ????????????????? an ????????????????????? n
  ???????? an ????????????????????????????
• Example
• ?????????? N 1 2 3 4 5

S 2 4 6 8 10
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Sequences

• ?????????????? an ????????(sequence) ???
  ??????????????????????????????????????????????
• ???????? ???? ??????????????????????????????????
  2, 4, 6, 8, 10, ?????????????????? an ??????
  an 2n ?????? n?1
• ????????????????????????? ???????????????? a1,
  a2, ??? an ???????? ????
• ??????????????? an a1, a2, , ?????? (?n?1)
  an f(n) 1/n
• ??????? ???????? an 1, 1/2, 1/3,

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The Formula Game
????????????????????????? a1, a2, a3,
????????????????????

• 1, 3, 5, 7, 9,

an 2n 1
-1, 1, -1, 1, -1,
an (-1)n
2, 5, 10, 17, 26,
an n2 1
0.25, 0.5, 0.75, 1, 1.25
an 0.25n
3, 9, 27, 81, 243,
an 3n
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Recognizing Sequences

• ??????????????????????????????????????????????
• ?????????????????????????????????????????????????
• ?????????????????????????????????
• ???????? ???? ??????????????? a1, a2, a3,
  ????????????????
• 1,2,3,4,
• 1,3,5,7,9,
• 0,3,8,15,24,...

an n
5
an 2n-1
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an n2-1
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Arithmetic Progression

• ????????????(Arithmetic progression)
  ??????????????????????? a, ad, a2d, a3d, ,
  and ?????? a, r?R
• ????? a ???????????????(initial term) ???????? d
  ???????????????(common difference)
• ???? -1, 3, 7, 11, 15,... a -1 d 4
• 7, 4, 1, -2, -5,... a 7 d -3
• 1, 3, 5, 7, 9,... a 1 d 2
• 5, 11, 17, 23, 29... a 5 d 6
• ???????? bn ????????????????? b1, b2, b3,
  ?????????????????

bn 4n-5
bn 10-3n
bn 2n-1
bn 6n-1
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Geometric Progression

• ?????????????(geometric progression)
  ??????????????????????? a, ar, ar2, ar3, , arn
  ?????? a, r?R ????? a ???????????????(initial
  term) ???????? r ????????????????(common ratio)
• ???? -1, 1, -1, 1,... a -1 r -1
• 1, -1, 1, -1,... a 1 r -1
• 10, 50, 250, 1250,... a 10 r 5
• 2, 2/3, 2/9, 2/27,... a 2 r 1/3
• ???????? bn ????????????????? b1, b2, b3,
  ?????????????????

bn (-1)n
bn (-1)n1
bn 2(5n)
bn 6(1/3)n
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Summations
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Summation Notation

• ?????????? an, ???????????????????????(lower
  bound) j?0, ????????????????????????(upper bound)
  k?j, ??????????????? an ??? j ??? k
  ??????????????
• ????????????? i ?????????????????

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Simple Summation Example

• ???????????????????????(predicate)????????????????
  ?????? ??????????????????? ????

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Summation Examples

• ????????????????????????? 1000 ???????????? an
  ?????????? ann2 ?????? n 1, 2, 3, ?

??????????????
?????????? ?

• ???????? 1 2 3 4 5 6 21

?????????? ?
??????????? ??????????????????????????????????????
??????
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Eulers Trick ???????????????????????

• ???????????? ????????12(n/2)((n/2)1)(n-
  1)n
• ???????????????????????? n/2 ???
  ?????????????????????????? n1,
  ?????????????????????????????????? (n/2)(n1)
• ???????????????????????
  ?????? ???????? ????

n1

n1
n1
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Summation Manipulations

• ??????????????????? ????

(????????????????????)
)
(
(???????????)
(??????????????)
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More Summation Manipulations

• ??????????????????? ????? ????

(????????????)
ç
(????????????)
1
)
(
(???????????)
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Example

• ??????????
• 51121- 101140 1085

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Example Arithmetic Progression

• ????????????? ??????????????????????????????? a,
  ad, a2d, a3d, , and ?????? a, r?R ??? n?0
• ?????????????????????????????? ???????????

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Example Geometric Progression

• ?????????????? ??????????????????????????????? a,
  ar, ar2, ar3, , arn ?????? a,r?R ??? n?0
• ?????????????????????????????? ???????????
• ???????????????????????????????????? ???

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Some Shortcut Expressions

??????????????(Geometric series)
?????????????(Eulers trick)
??????????????(Quadratic series)
??????????????(Cubic series)
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Using the Shortcuts

• ???????? ???? ??????? .
• ?????????????????
• ????????????????????
• ???????????????????
• ??????????

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Nested Summations

• ???????? ????????????????????
• ????????

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MathematicalInduction
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Mathematical Induction

• ???????????????????????????(mathematical
  induction) ???????????????????????????????????????
  ??????????????????????????????????????????????????
  ?????????????????????????(???????????????????????
  ???????????? ?n P(n) ??????????? ????? n?N)
• ??????????????????????????? P(n)
  ??????????????????????? P(n) ????????????
  ???????????????????? n ????????????????
• ??????? P(0) ???????? (?????????????? Basis
  step)
• ??????? ??? P(n) ???????????? P(n 1) ????????
  ???????????? n?N (????????????? Inductive
  step)
• ??????? P(n) ???????????? ???????????? n?N
  (??????????? Conclusion)

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Example 1

• ???????? ???? ???????????? ?n ? 1 P(n) ??????
  
• P(n) ?????????????????????????????? n ??????
  ???????????? n2

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Example 1

• ??????????????????????????????????????????? n
  ?????? ???????????? n2 ????????????????????
• ??????????????????????????????????
• Basis step ??? n1 ????????????????????????????
  ?? 1 ?????? ???????????? 12 (????)

P(n)
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Example 1

• Inductive step ?????????? ?n?1 P(n)?P(n1)
• ???????? n?1, ????? P(n) ???????? ?????????????
  P(n1) ????????????
• ??????? ??????????????????????????????
  ???????????????????????????????????????? n
  ?????? ???????????? n2

????????????????? P(n)????????
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Example 2

• ????????? ?????? n ???????????????????????????????
  ???????????
• 1222232n 2n1 1
• ??? P(n) ???????????????????????????????????????
• Basis step P(0) ???????? ????? 20 1 21-1
• Inductive step ???????? P(n) ????????
• ??????????? P(n1) ????????????

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Example 2 - Inductive Step

• P(n) ???????? ??????????? 12222n 2n1 1
• ???????????????????? 12222n1 2(n1)1 1
• 12222n1 12222n 2n1
• (2n1 1) 2n1
• 2.2n1 1
• 2n2 1
• ????????????????????????????????????????????
  ????????????? 1222232n 2n1 1 ???? ?

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Example 3

• ???????? ?????????????????? n 1 ????????
• Proof Basis step ????? n 1
  ????????????????????????
• Inductive step ???????????????????????????? k
  ??????? ????????
• ???????????????????????????????????????? k1 ????

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Example 3

• ??????? ????????,
• ??????? ????????????????????? k1
  ?????????????????????????? inductive step
• ????????????????? P(n) ????????????
  ?????????????? n?N ???????
• 1 2 n n (n 1)/2 ??????????????????????
  n?N ?

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