Sequence – a function whose domain is positive integers. Section 9.1 – Sequences.

9
Sequence – a function whose domain is positive integers. Sequence Notation ( ) =2 + 1 ( ) , ( ) ,h ( ) , , ( ) = 2 − 3 + 7 h ( ) = + 6 2 + 2 + 3 = 2 + 1 = 2 − 3 + 7 = + 6 2 + 2 + 3 Section 9.1 – Sequences

Transcript of Sequence – a function whose domain is positive integers. Section 9.1 – Sequences.

Page 1: Sequence – a function whose domain is positive integers. Section 9.1 – Sequences.

Sequence – a function whose domain is positive integers.

Sequence Notation

𝑓 (𝑥 )=2𝑥+1

𝑓 (𝑥 ) ,𝑔 (𝑥 ) , h(𝑥 ) 𝑎𝑛 ,𝑏𝑖 ,𝑐𝑘

𝑔 (𝑥 )=𝑥2−3 𝑥+7

h (𝑥 )= 𝑥+6

𝑥2+2 𝑥+3

𝑎𝑛=2𝑛+1

𝑏𝑖=𝑖2−3 𝑖+7

𝑐𝑘=𝑘+6

𝑘2+2𝑘+3

Section 9.1 – Sequences

Page 2: Sequence – a function whose domain is positive integers. Section 9.1 – Sequences.

Examples𝑾𝒓𝒊𝒕𝒆𝒕𝒉𝒆 𝒇𝒊𝒓𝒔𝒕 𝒇𝒐𝒖𝒓 𝒕𝒆𝒓𝒎𝒔 𝒇𝒐𝒓 {𝒂𝒌 }=𝒂𝒌=𝟐𝒌 .

2 (1 ) ,2 (2 ) ,2 (3 ) ,2(4)

2 ,4 ,6 ,8

Sequence Notation

𝑾𝒓𝒊𝒕𝒆𝒕𝒉𝒆 𝒇𝒊𝒓𝒔𝒕𝒕𝒉𝒓𝒆𝒆𝒕𝒆𝒓𝒎𝒔 𝒇𝒐𝒓 {𝒃𝒊 }=𝒃𝒊=𝒊𝟐

(𝟏 )𝟐 , (𝟐 )𝟐 , (𝟑 )𝟐

Section 9.1 – Sequences

𝟏 ,𝟒 ,𝟗

Page 3: Sequence – a function whose domain is positive integers. Section 9.1 – Sequences.

Sigma Notation – A mathematical notation that represents the sum of many terms using a formula.

Section 9.1 – Sequences

Page 4: Sequence – a function whose domain is positive integers. Section 9.1 – Sequences.

Examples

∑𝒏=𝟏

𝟒

𝟐𝒏

2 (1 )+2 (2 )+2 (3 )+2(4)

2+4+6+8

2+2+2+2+2+2

12

Sigma Notation

20

∑𝒌=𝟏

𝟔

𝟐

∑𝒊=𝟏

𝟑

(𝒊¿¿𝟐)¿

)

𝟏+𝟒+𝟗𝟏𝟒

Section 9.1 – Sequences

Page 5: Sequence – a function whose domain is positive integers. Section 9.1 – Sequences.

Express the sums in sigma notation.

∑𝒊=𝟏

𝟗𝟖

𝒊

1+2+3+4+…+98

1+12+

13+

14+…+

170

Sigma Notation

∑𝒌=𝟏

𝟕𝟎 𝟏𝒌

1−2+3−4+…−98

∑𝒊=𝟏

𝟗𝟖

(−𝟏)𝒊+𝟏𝒊

1−14+

19−

116

+…−149

∑𝒊=𝟏

𝟕

(−𝟏)𝒊+𝟏 𝟏𝒊𝟐

Section 9.1 – Sequences

Page 6: Sequence – a function whose domain is positive integers. Section 9.1 – Sequences.

Linearity of Sigma

∑𝒊=𝟏

𝒏

𝒄 𝒂𝒊=𝒄∑𝒊=𝟏

𝒏

𝒂𝒊

Sigma Notation

∑𝒌=𝟏

𝒏

(𝟑𝒌𝟐+𝟐𝒌−𝟕)

∑𝒊=𝟏

𝒏

(𝒂𝒊±𝒃𝒊)=¿∑𝒊=𝟏

𝒏

𝒂𝒊±∑𝒊=𝟏

𝒏

𝒃𝒊 ¿

∑𝒌=𝟏

𝒏

𝟑𝒌𝟐+∑𝒌=𝟏

𝒏

𝟐𝒌−∑𝒌=𝟏

𝒏

𝟕

Example

𝟑∑𝒌=𝟏

𝒏

𝒌𝟐+𝟐∑𝒌=𝟏

𝒏

𝒌−∑𝒌=𝟏

𝒏

𝟕→

Section 9.1 – Sequences

Page 7: Sequence – a function whose domain is positive integers. Section 9.1 – Sequences.

∑𝒌=𝟏

𝒏

𝒄=𝒏 ∙𝒄

Summation Rules

∑𝒌=𝟏

𝒏

𝒌𝟐=𝒏(𝒏+𝟏)(𝟐𝒏+𝟏)

𝟔

∑𝒌=𝟏

𝒏

𝒌=𝒏(𝒏+𝟏)

𝟐

∑𝒌=𝟏

𝒏

𝒌𝟑=¿(𝒏(𝒏+𝟏)𝟐 )

𝟐

¿

Section 9.1 – Sequences

Page 8: Sequence – a function whose domain is positive integers. Section 9.1 – Sequences.

∑𝒌=𝟏

𝟓𝟏

𝟒=¿¿

Summation Rules Examples

∑𝒌=𝟏

𝟏𝟓

𝒌𝟐=¿¿

∑𝒌=𝟏

𝟑𝟐

𝒌=¿¿

∑𝒌=𝟏

𝟗

𝒌𝟑=¿¿

𝟓𝟏 ∙𝟒=¿𝟐𝟎𝟒

𝟑𝟐(𝟑𝟐+𝟏)𝟐

=¿𝟓𝟐𝟖

𝟏𝟓(𝟏𝟓+𝟏)(𝟐 ∙𝟏𝟓+𝟏)𝟔

=¿𝟏𝟐𝟒𝟎

(𝟗(𝟗+𝟏)𝟐 )

𝟐

=¿𝟐𝟎𝟐𝟓

𝒔𝒖𝒎(𝒔𝒆𝒒 (𝟒 , 𝒙 ,𝟏 ,𝟓𝟒 ,𝟏 ))

𝒔𝒖𝒎(𝒔𝒆𝒒 (𝒙 ,𝒙 ,𝟏 ,𝟑𝟐 ,𝟏 ))

𝒔𝒖𝒎(𝒔𝒆𝒒 ( 𝒙𝟐 ,𝒙 ,𝟏 ,𝟏𝟓 ,𝟏))

𝒔𝒖𝒎(𝒔𝒆𝒒 ( 𝒙𝟑 ,𝒙 ,𝟏 ,𝟗 ,𝟏))

Section 9.1 – Sequences

Page 9: Sequence – a function whose domain is positive integers. Section 9.1 – Sequences.

∑𝒌=𝟏

𝟏𝟓

𝟒𝒌𝟐−𝟖𝒌+𝟔

Summation Rules Examples

𝟖∑𝒌=𝟏

𝟏𝟓

𝒌+¿¿

𝟏𝟓 ∙𝟔=¿

𝟗𝟎=¿

𝟖(𝟏𝟓 (𝟏𝟓+𝟏)𝟐 )+¿

𝟗𝟔𝟎+¿

𝟒 (𝟏𝟓 (𝟏𝟓+𝟏)(𝟐∙𝟏𝟓+𝟏)𝟔 )−𝟒𝟗𝟔𝟎−

𝒔𝒖𝒎 (𝒔𝒆𝒒 (𝟒 𝒙𝟐−𝟖 𝒙+𝟔 , 𝒙 ,𝟏 ,𝟏𝟓 ,𝟏))=¿

Section 9.1 – Sequences

𝟒∑𝒌=𝟏

𝟏𝟓

𝒌𝟐− ∑𝒌=𝟏

𝟏𝟓

𝟔

𝟒𝟎𝟗𝟎

𝟒𝟎𝟗𝟎


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