‫‪MST121‬المادة‪:‬‬
‫الدكتور‪:‬كمال الهادي عبد الرحمن‬
A1:SEQUENCES
• A sequence is an ordered list of numbersfinite or infinite-called terms.
• Notation: The nth term of a sequence is
denoted by an. The subscript or suffix n
indicates its position. The sequence itself
is either denoted by a or (an).
• Ex1: a=(an)=(1,4,9,16,25): finite
• Ex2: b=(bn)=(1,3,5,7,9,…) infinite
• Ex3: c=(cn)=(3,-2, 2,3,2,1)
Graph of a sequence: plotted by taking
ordered pairs (n,an) for n in the domain of
definition
an
.
.
.
.
n
1
2
3
4
• Describing sequences:
• A closed form of a sequence is a rule or
formula giving nth term an in term of n.
• Eg in Ex1: an=n2, n=1,2,3,4,5… perfect
squares.
in Ex2:bn=2n+1,n=0,1,2,3,…
Or bn=2n-1 n=1,2,3 odd numbers.
• Convention: first term is usually taken as
a1(starting from n=1).
• Sometimes we start from n=0, i.e. first
term in a0.
Examples: dn=1/n(n-1),n=2,3,…
un=(-1)n1/n, n=1,2,3,…
.
• A recurrence relation: gives any term of
the sequence in terms of the preceding
term. (a first order recurrence relation).
e.g. in Ex2: bn=bn-1+2, n=1,2,3,…
A recurrence system consists of a
recurrence relation, the first term and an
index range.
• Arithmetic sequences: are sequences
where the difference between any two
consecutive terms is constant-called the
common difference and denoted by d.
(Also called arithmetic progression)
Example: Ex2: bn=bn-1+2, here d=2.
• Arithmetic sequence: with first term a and
difference d is given by the recurrence
system:
x1=a,xn+1= xn+d,n=1,2,3….
• Parameters of the sequence are its first
term a and difference d.
• Closed from of an arithmetic sequence is
• x1=a,xn=a+(n-1)d,n=1,2,3…
or alternatively
x0 =a,xn=a+nd,n=0,1,2…
7
• GEOMETRIC SEQUENCES: are
sequence where the ratio between any
two consecutive terms is constant- called
the common ratio and denoted by r,(also
called geometric progression).
• Parameters are:first term a,common ratio r
• Recurrence relation:
x1=a, xn+1=rxn, n=1,2,3,..
Or x0=a, xn+1=rxn, n=0,1,2,..
Example: xn+1=2xn, n=1,2,…
If x1=2
If x1=3
(xn)=(2,4,8,16…)
(xn)=(3,6,12,24,..)
Closed from of a geometric sequence:
x1 =a, xn =arn-1 , n=1,2,3,…
OR
x0=a,xn=arn, n=0,1,2,3,…
• Sum Sn of first n terms of the
geometric sequence xn=rn and first
term =1:
Sn= 1+r+r2+…+rn-1 .
rsn= r+r2+.. +rn-1+rn
rsn—Sn=Sn(r-1)= rn-1
:. Sn = (rn-1 )/(r-1)
or (1-rn ) /(1-r)
,
r≠1
• Linear Recurrence Sequences :
each term is given as a linear function of
the preceding term.
• xn+1 = r xn +d , r , d constants , n = 1,2,3,…
• Parameters are : first term a , ratio r and
difference d.
• They generalize both arithmetic sequences and
geometric sequence (special cases ).
• (1) r =1
xn+1 =xn+d (arithmetic sequence )
(2) d = 0
xn+1 = rxn (geometric sequence).
• Recurrence system :
x1 = a , xn+1 = rxn +d , n = 1,2,3,…
or x0 = a , xn+1 = rxn +d , n = 0,1,2,3,..
• Closed form is
x1 = a , xn = (a+ d/(r-1) ) rn-1 – d/(r-1) ,n=1,2,3,…
or x0 = a , xn = (a+ d/(r-1) ) rn – d/(r-1),n=0,1,2…
• Long term behaviour of sequences :
For a sequence xn : as n → ∞ if xn → ℓ(i.e.
xn become arbitrarily close to ℓ ) then we
describe the long term ( or long run )
behaviour of the sequence by : xn → ℓ
as n → ∞.equivalently written lim xn =ℓ .
n
∞
Examples :
1) xn = k , constant : xn → k as n → ∞
2) xn = 1/n ,
xn → 0 as n → ∞
3) xn = n ,
xn → ∞ as n→ ∞
• For k >0
4) xn = nk
xn
-k , x
5) xn= 1=
n
n
k
n
Rules : if xn
(1) (xn ± yn )
(2) ( k xn )
(3) (xn. yn)
(4) (xn /yn)
∞ as n
0 as n
∞
∞
ℓ , yn
m as n
∞ then.
( ℓ ± m ) as n ∞.
kℓ
as n
∞.
ℓ. m as n
∞
ℓ / m m≠0 as n ∞
•
•
Examples : if an in a rational function of n:
write an=P /Q ,P,Q polynomials in n.
I If degree(P) > degree(Q) then an →∞ as n→∞
II If degree (P) < degree (Q) then an →0 as n→∞
III If degree (P)= degree (Q) then
an → coefficient of highest power in P÷
coefficient of highest power in Q as n → ∞
Long term behaviour of rn : (Table)
Range of r
r >1
r=1
-1 < r < 1
Behaviour of rn
rn →∞
as n →∞
rn →1
as n→∞
rn →0
as n →∞
r = -1
rn alternates between
as n →∞
1and-1 ,
r < -1
rn is unbounded and
alternates in sign as n→∞
A special limit
• an=(1+1/n)n→e ~ 2. 718.. as n →∞.
• bn = (1+x/n)n → ex
as n →∞.
Examble : ( 1- 5/n )n → e-5 as n →∞
• Sequences and Models : (A1 pp 36 -38 )
• Used Arithmetic sequence to model the book
and oil sequences.
• Used Geometric sequence to model the saving
account and bouncing ball sequences.
• Used linear recurrence sequence to model the
deer population and mortgage sequences.