Date
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Activities
/
Objectives
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Images
/
Home
work
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Assignments
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Remarks
/
Images
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References / Links
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Wednesday
March 9
11:25 -12:05
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- Registration sheets
- Introduction to this non-elective course
- Review of Arithmetic
and
Geometric
sequences : study of the main definition and
formulas.
- un = a + n.r ; vn = a.qn
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Comparison
of
the
arithmetic
and
geometric
means
of
2
numbers
:
a
geometric
illustration.
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Registration sheets
to be returned with a picture
on Friday 11th
- mean = 平均 [píng
jūn]
- sequence,
series of numbers
= 数列 [shù liè]
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Registration sheet
Arithmetic &
Geometric Sequences Memo
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Friday
March 11
11:25
-12:05
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- Call for the registration sheets
- Construction modes of a sequence :
- randomly
- function : un = f(n)
- recursion : un+1 = f(un)
;
u0= a
- Sum : Sn= u1+
u2+ u3+...+ un.
- double recursion :
un+2 = f(un;un+1) ; u0= a ; u1= b
- mixed ...
- Study topics of a
sequence :
- Formula ?
- Graphic Construction ?
- Monotony ?
- Boudaries ?
- Limit ?
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Examples :
- {6 ; 4 ; 12 ; 13 ...} Randomly
-
f(x) = -.5 x + 2
un = f(n) = -.5 n + 2
(Arithmetic sequence)
(Homographic
sequence)
- f(x) =
- 0.5 x + 2 ; un+1
= f(un)
un+1
= - 0.5 un + 2 ; u0
= 0
- Sn= a + aq + aq2 + ... + aqn
(Geometric
Series)
- Fn+2
= Fn + Fn+1 (Fibonacci)
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- for any n : -2 ≤ un ≤ 4
- for any n : un ≤ un+1
- lim un = 4
Construction
of
a
sequence
defined
by recursion whith an elementary function :
f(x) = .5 x + 2
un+1 = f(un) = .5 un+
2
Exercise # 1.1
vn
= un -
4
(vn)
is a geometric sequence
(to be checked)
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Exercise # 1.1
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March 16
11:25 -12:05
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- Collect more Registration sheets
- Check the Exercise # 1.1
- Review of Limits of a
Geometric Sequence
vn = v0. qn
- q > 1
1.
v0 > 0
==>
lim
vn=
+
∞
2.
v0 < 0 ==> lim vn= - ∞
- 0 < q < 1
==> lim vn=
0+
- q = 1 ==>
lim vn=
v0
- -1 < q <0
==>
lim
vn=
0
- q =-1
==> No limit
- q < -1
==> No limit
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Home Work :
Complete Exercise # 1.2
Construction
of
a
sequence
defined
by recursion whith an elementary function :
f(x) = -.5 x + 2
un+1 = f(un)
=
-.5
un+
2
Exercise # 1.2
vn
= un -
4/3
(vn)
is a geometric sequence
(to be checked)
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Exercise # 1.2
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Friday
March 18
11:25
-12:05
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- Collect last Registration sheets
- Check Exercise # 1.2
- Review of the definition of a geometric
sequence.
- Handout Exercise # 2.1
- Introduction to the general definition of the limit
of a
sequence (see Ex. 2.1) :
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Non recursive sequence
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March 23
11:25 -12:05
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- Comparison between recursive and non recursive Sequences
defined by the same function.
- Use of an auxiliary Geometric
sequence to find the limit.
 v0 = 4
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v0 = 4
Geom.
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Exercise # 2.2
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Friday
March 25
11:25
-12:05
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- Collect exercise # 2.2
- Return Exercises 1.2 & 2.1
- Handout ANSWERS of Ex # 2.1 & 2.2
- Handout ANSWERS of Ex # 1.1 & 1.2
- New exercise 2.3 on recursive sequence :
- Use of an auxiliary Arithmetic
sequence to find the limit.
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 (Arithmetic) |
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Ex. 1.1 & 1.2 ANSWERS
Ex. 2.1 &
2.2 ANSWERS
Exercise # 2.3
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March 30
11:25 -12:05
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- Return Exercise 2.2
- Collect Exercise # 2.3
- Handout Answers Ex.
2.3
- Study of the special recursive sequences defined by
f(x) = r. x(a-x)
un+1 = f(un) ; 0 ≤ u0 ≤
a
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Exercise # 2.3
ANSWERS
Ex. # 2.3
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