Introduction to Finance 6 - Arithmetic and geometric progressions PDF

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Arithmetic and geometric sequences
Lecture by Sir Kevin Albertson...

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IIntroduction t d ti to t Finance Fi 6 – Arithmetic and geometric progressions Kevin Albertson [email protected]

 

Mathematical Economics  (5L4Z0009)

Present Values

 Finite progressions  Infinite progressions



  ! "!#"$$$

 

 Arithmetic progressions  Geometric Progressions

Arithmetic Progressions



  ! "!#"$$$

 

 An arithmetic progression is a sequence of terms in which each term is obtained by adding a constant factor to the preceding term 2 4 6 8 10 … 3 6 9 12 15 … 10 9 8 7 6 5 4 3 2 1 …  The initial term is denoted , and the increment .  The series is denoted  + +2 +3 +4 and so on.  The th term is  + (–1)

Geometric Progressions



  ! "!#"$$$

 

 A geometric progression is a sequence of terms in which each term is obtained by multiplying the preceding term by a constant factor eg e.g. 1 2 4 8 16 32 … 32 8 2 ½ …  Let us denote the initial term in the sequence  and the common factor   The series is described by:   2 3 4 5 6 …

Geometric Progressions



  ! "!#"$$$

 

 Let us denote the initial term in the sequence  and the common factor   The series is described by:   2 3 4 5 6 …  The 5thh term of the sequence is 4 the 20th term of the sequence is 19 the th term of the sequence is –1

 Suppose we have a progression with a finite number of terms, for example 1 2 4 8 16 32 64 128  We wish to determine the sum of the elements of this series  Let the sum of the first  terms in a geometric progression be denoted   =  +  + 2 + 3 … + –1 

  ! "!#"$$$

 

Geometric Progressions Sum of progressions of finite order

 Let the sum of the first  terms in a geometric progression be denoted   =  +  + 2 + 3 … + –1



  ! "!#"$$$

 

Geometric Progressions Sum of progressions of finite order

 Let the sum of the first  terms in a geometric progression be denoted   =  +  + 2 + 3 … + –1  =  + 2 + 3 … + –11 + 



  ! "!#"$$$

 

Geometric Progressions Sum of progressions of finite order

 Let the sum of the first  terms in a geometric progression be denoted   =  +  + 2 + 3 … + –1  =  + 2 + 3 … + –11 +   –  =  – 



  ! "!#"$$$

 

Geometric Progressions Sum of progressions of finite order

 Let the sum of the first  terms in a geometric progression be denoted   =  +  + 2 + 3 … + –1  =  + 2 + 3 … + –11 +   –  =  –   (1 – ) =  –  

  ! "!#"$$$

 

Geometric Progressions Sum of progressions of finite order

 Let the sum of the first  terms in a geometric progression be denoted 

 (1 −   )  = 1−   NB This formula is invalid when  = 1



  ! "!#"$$$

 

Geometric Progressions Sum of progressions of finite order

 Consider the series: 48 12 3 ¾ …  = 48,  = ¼  The 6th element of the series is 48 × (¼)5 = 0%046875  The sum of the first 6 elements of this series is

48(1 − ¼ 6 ) = 63·984375 6 = 1− ¼ 

  ! "!#"$$$

 

Geometric Progressions Sum of progressions of finite order

 Consider the series: 48 12 3 ¾ …  = 48,  = ¼  The 6th element of the series is 48 × (¼)5 = 0%046875  The sum of the first 10 elements of this series is

48(1 − ¼10 ) = 63·99993  10 = 1− ¼ 

  ! "!#"$$$

 

Geometric Progressions Sum of progressions of finite order

 Consider the series where  = 16,  = ¾    1 2 3 4 5



16 12 9 6%75 5%0625

16 28 37 43%75 48%8125

  ! "!#"$$$

 

Geometric Progressions Sum of progressions of infinite order



 = 16,  = ¾



  ! "!#"$$$

 

Geometric Progressions Sum of progressions of infinite order

 As  gets larger and larger,  gets closer and closer to 64  We write this  → 64 as  → ∞ or Limit() = 64 →∞



  ! "!#"$$$

 

Geometric Progressions Sum of progressions of infinite order



 = 16,  = 1¼



  ! "!#"$$$

 

Geometric Progressions Sum of progressions of infinite order

 Progressions of infinite order  Two cases  The sum of the series has a limit The progression is convergent  The sum of the series has no limit The progression is divergent



  ! "!#"$$$

 

Geometric Progressions Sum of progressions of infinite order

 Progressions of infinite order  Two cases  The process is convergent when –1 <  < 1  The process is divergent when  < –1 or  > 1



  ! "!#"$$$

 

Geometric Progressions Sum of progressions of infinite order

 Convergent Progressions of infinite order  The sum  =  +  + 2 + … + ∞ where || < 1 is given by  Limit   =  →∞ 1−



  ! "!#"$$$

 

Geometric Progressions Sum of progressions of infinite order

 Consider the series where  = 16,  = ¾

Limit   =  →∞

16 16  = = 1 = 64 1 −  1 − 34 4

 Consider the series where  = 1,  = 0%9

Limit   =  →∞



 1 1 = = = 10 1 −  1 − 0·9 0·1

  ! "!#"$$$

 

Geometric Progressions Sum of progressions of infinite order

 ...