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Arithmetic Sequences

Geometric Sequences

Arithmetic Series

Geometric Series

Sigma Notation

Mathematical Induction

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 UNIT 10 : SEQUENCES AND SERIES

 LESSON 7: UNIT SUMMARY

 

Arithmetic Sequences:

A sequence such as 2, 3, 8, 13, is called an Arithmetic Sequence. These sequences have the following properties.

        Terms are denoted as t1 , t2 , t3 , referring to term1, term 2, term 3

        The difference between successive terms is constant. ie t2 t1 = t3 t2 = t4 t3 etc

        This difference is called the common difference and denoted using the letter d. Here d = 5.

        The first term is denoted using the letter a. Here a = -2.

        Successive terms are found by adding the common difference, d, to the preceding term. Hence t5 = 13 + 5 = 18 etc.

        The formula for the general term or nth term is tn = a + (n 1)d

        Arithmetic sequences are linear functions with domain the natural numbers N = {1, 2, 3, 4, 5, }

 

Text Box: a = t1 = first term
d = common difference
n = term number
tn = nth term or general term
 

 

 

 

 

 

 


Arithmetic Series:

 

Definition: The sum of the terms of an arithmetic sequence is an Arithmetic Series.

Text Box:

Text Box: n = the number of terms
t1 = a = the first term
tn = l = the last term
d = the common difference
Sn = the sum of the series
 

 

 

 

 

 

 

 


Example:

 

a = 255

d = -4

n = ?

tn = 3

Sn = ?

 

 

Geometric Sequences:

A sequence such as 2, 4, 8, 16, 32, is called a Geometric Sequence. These sequences have the following properties.

        Terms are denoted as t1 , t2 , t3 , referring to term1, term 2, term 3

        The ratio of any term to the term preceding is constant.

        This ratio is called the common ratio and denoted using the letter r. Here r = 2.

        The first term is denoted using the letter a. Here a = 2.

        Successive terms are found by multiplying a given term by the common ratio. Eg. t6 = 32 x 2 = 64 etc.

        The formula for the general term or nth term is tn = arn-1.

        Geometric sequences are exponential functions with domain the natural numbers N = {1, 2, 3, 4, 5 }

Text Box: a = t1 = first term
r = common ratio
n = term number
 

 

 

 

 

 


Geometric Series:

 

Definition: The sum of the terms of a Geometric sequence is a Geometric Series.

Text Box: n = the number of terms
t1 = a = the first term
r = the common ratio
Sn = the sum of the series

Text Box:
 

 

 

 

 


Example:

 

a = 4

r = 2

n = ?

tn = 1024

Sn = ?

 

 

 

Sigma Notation:

 

 

 

 

Mathematical Induction:

 

Axiom of Mathematical Induction:

If a certain hypothesis is true for a finite set of natural numbers S, we need to show that the set S equals N, the set of all natural numbers.

The following axiom of induction will assist us in this quest.

 

 

 

 

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