Home Arithmetic Sequences Geometric Sequences Arithmetic Series Geometric Series Sigma Notation Mathematical Induction Review&Test

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, …}

Arithmetic Series:

Definition: The sum of the terms of an arithmetic sequence is an Arithmetic 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 …}

Geometric Series:

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

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.