
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 t_{1} , t_{2} , t_{3 , }referring
_{ }to term1, term 2,
term 3 …
·
The difference between successive terms is constant. ie t_{2}
– t_{1} = t_{3} – t_{2} = t_{4} – t_{3}
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 t_{5} = 13 +
5 = 18 etc.
·
The
formula for the general term or nth term is t_{n }= 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 = ? t_{n} = 3 S_{n} = ?
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 t_{1} , t_{2} , t_{3 , }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.
t_{6} = 32 x 2 = 64 etc.
·
The formula
for the general term or nth term is
t_{n }= ar^{n1}.
·
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 = ? t_{n} = 1024 S_{n} = ?
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.