Eng.sherif yehia
0101560322
Block A1 & B1
Sequences
Block A1
Introduction:
A Sequence is a list of Numbers May be Used to Represent a quantity
that's Varying Over Time Such as :1, 3, 5, 7 …
This Number is called “terms “of Sequence.
Where 1 =1rst term , 3 = 2nd term , 5 = 3rd term , 7 = 4th term and
so on….
Sequence in closed form
Any Formula defining a sequence terms of n is called a closed form
sequence.
S n = n2
Like
(n = 1 , 2 . 3 . …)
Example (1)
Write down the first five terms, and the 100 th term of each of the
following.
(a)
an = 7n
( n = 1 , 2 , 3 …… )
Answer
Answer
a1 = 7 ، a2 = 14 ، a3 = 21 ، a4 = 28
، a5 = 35
a100 = 700
unit A1 & B1
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The sequence
Eng.sherif yehia
0101560322
( n = 1 , 2 , 3 …… )
bn = ( - 1 ) n+1
(b)
Answer
Answer
، b2 = -1 ،
b1 = (- 1)2 = 1
b4 = - 1
b3 = 1
b5 = 1
Example (2)
(2)
Example
( a ) Specify a Sequence in closed Form whose First four terms are
1,8,27,64 Then get the 10th term in your Sequence .
Answer
Answer
xn = n3 ( n = 1,2,3, 4 , ……)
as
x1=13 =1 ، x2 =23 = 8 ، x3 = 33 = 27 ، x4 = 4 3 = 64
Recurrence Relation
System
For formula
1)
( n = 1 , 2 , 3 ….. )
an = 7n
.: a1 = 7
، a2 = 14
، a3 = 21
، a4 = 28
Which Means
،
a2 = a 1 + 7
Recurrence System
2)
b n = 2n
.: b1 = 2
،
a3 = a2 + 7
a4 = a3 + 7
X n + 1 = xn + 7
( n = 1 , 2 , 3 ….. )
، b2 = 4
، b3 = 8
، b4 = 16
Means That
b2 = 2b1
،
b3 = 2b2
،
b4= 2b3
X .:n + 1 = 2 xn
unit A1 & B1
- 66 -
The sequence
Eng.sherif yehia
0101560322
Example (3)
(3)
Example
Write down the first four terms of each of the following sequences.
(a) a1 = 0 , a n + 1 = 2an + 1 ( n= 1,2,3,4…)
Answer
Answer
a1 = 0
a2 = 2a1 + 1 = 0 + 1 = 1 ,
then a2 = 1
a3 = 2 a2 +1 = 2(1) + 1 = 3 , then a3 = 3
a4 =2a3 + 1 = 2(3) + 1 = 7 , then a4 = 7
(b) c0 = 2 , c n + 1 = 3cn
( n = 0,1,2,3…)
Answer
Answer
c0 = 2
c1 = 3c0 = 3(2) = 6 , then c1= 6
c2 = 3c1 = 3(6) =18 , then c2= 18
c3 = 3c2 = 3(18) = 54 , then c3= 54
c4 = 3c3 = 3(54) = 162 , then c4= 162
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unit A1 & B1
- 67 -
The sequence
Eng.sherif yehia
0101560322
Arithmetic sequence (A.S.)
The sequence (xn) is called arithmetic sequence if :
x n + 1 - x n = constant number
This means that any term of the (A.S) – previous term = constant
number which is called the common difference denoted by "d".
Example (1)
(1)
Example
Which of the following sequences is an arithmetic sequence and
which is not?
(a)
X1 = -1
X n + 1 = Xn + 1
(b)
Y1 = 2
Y n + 1 = -Yn + 1
(c)
Z0 = 1
Z n + 1 = Zn - 1
A closed form for A.S
For
x n + 1 = xn + d
x1 = a
Then
and For x0 = a
Then
( n = 1,2,3….)
xn = a + (n-1)d
x n + 1 = xn + d
( n = 0,2,3….)
xn = a + nd
Where (a) is the first term, (d) is the constant difference
unit A1 & B1
- 68 -
The sequence
Eng.sherif yehia
0101560322
Geometric sequence (G.S)
For:
X1 = a
X n + 1 = r xn
Where r is Constant Ratio
r = (X n + 1 ) / Xn
Example (1)
(1)
Example
Which of the Following is Geometric Sequence.
(1)
X-1 = -1
X n + 1 = 3Xn
(2)
y0 = 2
y n + 1 = - 0.9Yn
(3)
Z1 = 2
Z n + 1 = - Zn + 1
A closed form for G.S
For:
X1 = a
Then
X n + 1 = r xn
Xn = a(r)n-1
X0 = a
Then
unit A1 & B1
X n + 1 = r xn
Xn = a(r)n
- 69 -
The sequence
Eng.sherif yehia
0101560322
Linear recurrence Sequence
Arithmetic ‫شـاملة الـــ‬
Geometric
Such as:
For:
X1 = a
X n + 1 = rX n + d
Xn = (a + d )r n-1 - d
r-1
And for:
X0 = a
r-1
X n +1 = r X n + d
Xn = (a + d )r n - d
r-1
r-1
Example
Example(2)
(1)
For each of the Following of sequence get the First Term and
Ratio and recurrence System.
(a)
1 , ½ , ¼ , ⅛ , ….
Answer
Answer
a =1 ،
Recurrence System:
(b)
r = ½
X n +1 = ½ X n
2 , -2 , 2 , -2 , ….
Answer
Answer
a1 = 2
Recurrence system
unit A1 & B1
r = -1
X n + 1 = -Xn
- 70 -
The sequence
Eng.sherif yehia
0101560322
long Term Run ‫مهم جدا‬
(Converges)
Examples
Examples:
What is the long term of :
(a)
Xn
=
2n + 1
n = (1,2,3)
Answer
Answer
As (n) becomes Large , as 2 n also becomes larger which makes the
sequence Xn unbounded so it doesn’t converge .
Lim Xn = ∞
n
(b)
∞
Xn = 45 (- 1/3 )n – 1 + 3/4
( n = 1 , 2 , 3…)
4
Answer
Answer
as (n ) becomes Large , as (- 1/3 )n – 1 becomes Small and make the
sequence 45 (- 1/3 )n – 1 converges to zero , so the sequence Xn
4
Converges to 3/4 .
nn
(c)
Xn = (-1)n + 1 ∞
Li m Xn = 3/4
∞
( n = 1 , 2 , 3 .. )
Answer
Answer
as (n ) becomes Large , as (- 1) n
arbitrary Converges between
( -1 ,1) which means that Xn converges beteen (0,-2) .
Lim Xn = ( 0 – 2 )
∞
nn
unit A1 & B1
∞
- 71 -
The sequence
Eng.sherif yehia
0101560322
Example(2)
(2)
Example
State the long term behavior term of the sequences below and state
Weather Converges or not , if it does , state the limit
(a) an
= (0.6)n + 2(1.4) n
Answer
Answer
as (n) becomes large , the sequence (0.6)n converges to zero, then
the sequence an doesn't converge (unbounded).
(b)
an
=
5
17 - 3 (2.1) n
Answer
Answer
As (n) becomes large , as sequence (2.1)n becomes unbounded ,
which makes the sequence an converges to zero .
Lim Xn = 0
∞
n
(c) an
=
n²
∞
5 + 10n
Answer
Answer
We can make an
=
n
5/n + 10
for sequence (5/n), as (n) becomes large , as (5/n) converges to zero ,
then the sequence an converges to 1 n
which is not
convergent (unbounded).
10
∞
(d) an = 100
4 + 20(0.6)n
Answer
Answer
As (n) becomes large , (0.6)n become small which converges 20(0.6)n
converges to zero making the sequence an converges to (100/4 = 25)
Lim an = 25
n ∞
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unit A1 & B1
∞
The sequence