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4
3
2
1
Outside the high school program
/ Other sequences
10 exercises (including 4 corrected)
a
Fibonacci sequences and numbers
(7 exercices)
Jointly defined sequences (final grades)
(1 exercice)
A little further: homographic sequences
(3 exercices)
A
B
1
2
1.
Fib
onac
ci
se
quenc
es
and
numb
ers
E.590
The
con
tin
uum
h
yp
othesis
assumes
that
an
y
part
of
the
line
can
b
e
cut
indefinitely
and
that
eac
h
part
has
a
non-zero
length.
This
assumption
runs
coun
ter
to
the
concept
of
matter
as
the
mass
of
an
atom,
but
is
widely
used
in
quan
tum
theory
for
motion
phenomena.
Infinitesimal
calculus
is
the
branc
h
of
mathematics
that
stud-
ies
the
infinitely
far
and
the
infinitely
near.
(p
ar
adox
of
Zeno
of
Ele
a)
ˇThere
is
no
motion,
as
the
mobile
m
ust
reac
h
the
middle
of
its
path
b
efore
reac
hing
the
endı
This
parado
x
is
supp
osed
to
defeat
con
tin
uit
y
:
i.e.,
a
length
can
b
e
divided
indefinitely
.
Concept
opp
osite
of
discrete
(atomistic
the
ory)
E.1872
Consider
a
p
oin
t
X
mo
ving
from
A
to
B
describing
the
en
tire
segmen
t.
W
e
break
do
wn
the
p
oin
t’s
path
as
follo
ws
:
W
e’ll
sa
y
that
this
p
oin
t
arriv
es
at
the
first
stage,
when
it
has
tra
v
eled
half
of
[
AB
]
:
w
e’ll
note
X
1
this
p
oin
t.
It
will
b
e
at
the
second
stage
when
it
has
co
v
ered
half
of
the
remaining
path,
i.e.
half
of
[
X
1
B
]
:
this
p
oin
t
will
b
e
noted
X
2
.
The
third
step
will
b
e
half
[
X
2
B
]
:
half
the
distance
restan
t.
.
.
And
so
on.
Consider
X
0
as
the
starting
p
oin
t
A
.
1
Place
the
p
oin
ts
X
1
,
X
2
,
X
3
and
X
4
on
the
straigh
t
line
b
elo
w.
2
It
will
no
w
b
e
assumed
that
distance
AB
measures
1
me-
ter.
a
Giv
e
the
distance
X
0
X
1
,
X
1
X
2
,
X
2
X
3
et
X
3
X
4
.
b
By
extrap
olation,
giv
e
the
measuremen
ts
of
X
4
X
5
,
X
5
X
6
,
X
6
X
7
and
X
7
X
8
.
c
F
or
n
∈
N
,
mak
e
a
guess
as
to
the
distance
X
n
−
1
X
n
.
3
W
e
prop
ose
to
study
the
infinite
sum
of
terms
:
S
=
1
2
+
1
4
+
1
8
+
1
16
+
·
·
·
T
o
do
this,
for
n
∈
N
,
consider
the
sum
to
n
terms
:
S
n
=
1
2
+
1
4
+
1
8
+
·
·
·
+
1
2
n
a
W
rite,
without
calculating,
the
sums
S
3
,
S
4
and
S
5
.
b
Establish
the
form
ula
:
S
n
+1
=
1
2
+
1
2
·
S
n
c
Giv
e
a
reason
wh
y
the
n
um
b
ers
S
n
approac
h
the
v
alue
1
when
the
n
um
b
er
of
steps
b
ecomes
v
ery
large.
W
e’ll
sa
y
that
S
n
admits
a
limit
when
n
tends
to
+
∞
.
d
Noting
S
the
limit
v
alue
of
S
n
when
n
tends
to
+
∞
:
w
e’ll
note
lim
n
↦→
+
∞
S
n
=
S
.
By
passing
to
the
limit
of
the
form
ula
2
,
w
e
obtain
the
equalit
y
:
S
=
1
2
+
1
2
·
S
.
Find
the
v
alue
of
S
.
W
e’v
e
just
highligh
ted
one
of
the
prop
erties
of
the
con
tin
uum
h
yp
othesis
:
T
out
segmen
t
can
b
e
cut
in
to
an
infinite
n
um
b
er
of
segmen
ts
all
of
non-zero
length.
E.1873
Definition
:
(fr
om
L
e
Petit
R
ob
ert)
dic
hotom
y
:
Division,
binary
sub
division
(b
etwe
en
two
elements
that
ar
e
cle
arly
sep
ar
ate
d
and
opp
ose
d)
Assuming
the
thesis
of
con
tin
uit
y
,
an
y
ob
ject
is
infinitely
di-
visible.
W
e
kno
w
that
2
∈
[1
;
2]
b
ecause
:
1
2
<
2
2
<
2
2
T
w
o
n
um
b
ers
and
their
squares
are
arranged
in
the
same
order.
1
<
2
<
2
Implemen
tation
of
the
algorithm
:
A
t
step
0
(the
initial
step)
,
w
e
kno
w
that
2
is
con
tained
in
the
in
terv
al
[1
;
2]
T
o
mo
v
e
on
to
the
next
step,
w
e
consider
the
middle
of
this
in
terv
al
1
;
5
.
Then,
w
e
compare
1
;
5
and
√
2
to
deter-
mine
whether
√
2
is
con
tained
in
the
in
terv
al
[1
;
1
;
5]
or
the
in
terv
al
[1
;
5
;
2]
.
This
giv
es
us
a
new
in
terv
al
corresp
onding
to
step
1.
W
e
rep
eat
this
pro
cess
an
infinite
n
um
b
er
of
times.
Let
[
a
n
;
b
n
]
b
e
the
in
terv
al
corresp
onding
to
step
n
and
u
n
its
length.
1
a
Giv
e
the
v
alue
of
u
0
,
u
1
,
and
u
2
.
b
Place
p
oin
ts
a
0
,
b
0
,
a
1
,
b
2
,
a
3
,
and
b
4
on
the
graduated
line
b
elo
w.
2
a
Giv
e
the
v
alue
of
u
n
based
on
n
.
b
Giv
e
a
range
for
b
n
−
a
n
for
n
=
10
.
Deduce
the
accuracy
of
the
appro
ximation
of
2
b
y
u
10
3
What
happ
ens
to
the
v
alue
of
u
n
when
n
approac
hes
+
∞
?
(W
e
say
that
n
tends
towar
d
infinity)
.
That
is,
what
is
the
v
alue
of
lim
n
↦→
+
∞
u
n
?
h
ttps:/
/c
hingmath.fr
sacados/590
sacados/1872
A
B
sacados/1873
1
2
A
0
T
0
t
t
0
A
1
T
1
t
t
1
t
t
2
E.1874
A
c
hilles
and
the
tortoise
(Zeno
of
Ele
a’s
p
ar
adox)
Problem
:
if
the
tortoise
is
ahead
of
A
c
hilles,
A
c
hilles
will
nev
er
b
e
able
to
catc
h
up
with
it,
regardless
of
his
sp
eed
;
for
when
A
c
hilles
runs
to
reac
h
the
p
oin
t
from
whic
h
the
tortoise
started,
the
tortoise
mo
v
es
forw
ard
in
suc
h
a
w
a
y
that
A
c
hilles
can
nev
er
mak
e
up
the
difference.
Let
us
assume
that
the
tortoise
has
a
one-meter
lead,
that
A
c
hilles
runs
at
a
sp
eed
of
2
m
=
s
and
the
tortoise
at
a
sp
eed
of
1
m
=
s
.
Let
t
0
b
e
the
starting
time,
A
0
the
initial
p
osition
of
A
c
hilles,
and
T
0
that
of
the
tortoise.
t
1
is
the
momen
t
when
A
c
hilles
reac
hes
the
turtle’s
previous
p
osition.
Let
us
denote
this
p
osition
b
y
A
1
.
The
turtle
con-
tin
ues
to
mo
v
e
forw
ard
and
finds
itself
at
T
1
.
And
so
on.
Based
on
the
fact
that
A
c
hilles’
and
the
tortoise’s
paths
can
b
e
divided
in
to
as
man
y
segmen
ts
as
(c
ontinuity
hyp
othesis)
,
it
is
b
eliev
ed
that
A
c
hilles
will
nev
er
catc
h
up
with
the
tor-
toise,
as
there
will
alw
a
ys
b
e
a
p
ortion
of
the
path
left
for
it
to
tra
v
el.
Another
problem
:
what
is
the
total
length
of
all
these
small
segmen
ts
when
placed
end
to
end?
In
other
w
ords,
will
A
c
hilles
catc
h
up
with
the
tortoise
infinitely?
1
Place
p
oin
ts
A
2
and
T
2
on
the
last
graduated
line
2
Giv
e
the
v
alues
of
T
0
−
A
0
,
T
1
−
A
1
,
T
2
−
A
2
.
3
W
e
accept
the
fact
that,
at
an
y
rank
in
the
study
,
w
e
ha
v
e
:
u
n
=
T
n
−
A
n
=
1
2
n
.
W
e
will
calculate
ho
w
far
from
the
A
c
hilles
p
oin
t
the
par-
ticipan
ts
will
meet.
W
e
note
:
u
n
=
u
0
+
u
1
+
u
2
+
·
·
·
+
u
n
a
Expand
:
1
−
1
2
×
S
n
b
Deduce
:
S
n
=
1
−
1
2
n
+1
1
−
1
2
c
Giv
e
the
limit
of
S
n
when
n
tends
to
w
ards
infinit
y
,
whic
h
w
e
note
as
:
lim
n
↦→
+
∞
u
0
+
u
1
+
·
·
·
+
u
n
ou
lim
n
↦→
+
∞
+
∞
k
=0
u
n
.
Note
:
Zeno’s
parado
xes
w
ere
rep
orted
b
y
Aristotle
in
his
b
o
ok
on
ph
ysics
(Bo
ok
VI)
.
W
e
could
also
ha
v
e
considered
the
t
w
o
functions
that
as-
so
ciate
the
p
osition
of
A
c
hilles
and
the
tortoise
on
the
line
with
time
and
determined
the
p
oin
t
of
in
tersection.
E.1875
The
discon
tin
uit
y
(p
ar
adox
of
Zeno
of
Elia)
E.182
1
Consider
the
table
b
elo
w
:
n
0
1
2
3
4
5
6
7
8
9
10
11
F
n
0
1
1
2
3
5
8
13
21
34
55
89
Sho
w
that,
for
an
y
natural
in
teger
n
greater
than
2
,
the
n
um
b
ers
F
n
v
erify
the
follo
wing
relationship
:
F
n
=
F
n
−
1
+
F
n
−
2
The
n
um
b
ers
(
F
n
)
are
called
the
ˇ
n
um
b
ers
of
Fib
onacci
ı
;
an
y
sequence
of
n
um
b
ers
exhibiting
suc
h
a
relationship
will
b
e
said
to
b
e
of
ˇ
Fib
onacci
suites
ı.
More
precisely
,
the
sequence
studied
ab
o
v
e
is
defined
b
y
the
initial
conditions
:
F
0
=
0
;
F
1
=
1
It
is
clear
that
:
lim
n
↦→
+
∞
F
n
=
+
∞
.
W
e
will
no
w
study
the
ratio
of
t
w
o
consecutiv
e
terms
;
i.e.
w
e
will
study
the
v
alue
of
the
quotien
t
F
n
+1
F
n
when
n
b
ecomes
larger
and
larger
(i.e.
when
n
tends
towar
ds
+
∞
)
.
2
a
F
or
n
=
2
,
establish
the
follo
wing
equalit
y
:
F
n
+1
F
n
=
2
b
Complete
the
follo
wing
table
to
within
10
−
5
:
n
1
2
3
4
5
6
7
8
9
10
F
n
+1
F
n
3
Mak
e
a
conjecture
as
to
the
limit
of
the
sequence
(
F
n
)
and
the
golden
n
um
b
er
1+
5
2
.
W
e
will
no
w
sho
w
that
all
Fib
onacci
sequences
ha
v
e
the
ratio
of
its
consecutiv
e
terms
tending
to
w
ards
the
golden
n
um
b
er
:
4
Noting
R
n
the
v
alue
of
the
ratio
F
n
+1
F
n
,
sho
w
that
:
R
n
=
1
+
1
R
n
−
1
5
Assuming,
that
the
sequence
(
R
n
)
admits
a
finite
limit
r
when
n
tends
to
+
∞
,
deduce
that
r
v
erifies
the
relation
:
r
2
=
r
+
1
6
V
erify
that
1+
5
2
and
1
−
5
2
are
t
w
o
n
um
b
ers
v
erifying
this
equation.
7
Assuming
the
fact
that
a
second-degree
p
olynomial
can
only
cancel
in
at
most
t
w
o
v
alues,
sa
y
wh
y
the
sequence
of
ratios
F
n
+1
F
n
necessarily
tends
to
1+
5
2
2.
Jointly
define
d
se
quenc
es
(final
gr
ades)
h
ttps:/
/c
hingmath.fr
sacados/1874
A
0
T
0
t
t
0
A
1
T
1
t
t
1
t
t
2
sacados/1875
sacados/182
0
2
4
6
8
10
12
14
2
4
6
8
E.5900
W
e
define
the
sequences
u
n
and
v
n
on
the
set
N
of
natural
n
um
b
ers
b
y
:
u
0
=
0
;
v
0
=
1
;
u
n
+1
=
u
n
+
v
n
2
v
n
+1
=
u
n
+
2
·
v
n
3
The
aim
of
this
exercise
is
to
study
the
con
v
ergence
of
the
sequences
u
n
and
v
n
.
P
art
A
1
Calculate
u
1
and
v
1
.
2
Consider
the
function
f
extracted
from
an
algorithm
où
the
parameter
n
is
an
in
teger
greater
than
or
equal
to
1
:
Function
f(n)
u
←
0
v
←
1
For
k
varying
from
1
to
n
w
←
u
u
←
w+v
2
v
←
w+2
·
v
3
End
For
Return
(
u
;
v
)
a
Call
the
function
f
with
the
v
alue
n
=
2
as
its
argumen
t.
Cop
y
and
complete
the
table
b
elo
w
sho
wing
the
state
of
the
v
ariables
during
the
call
to
the
function
f
.
k
w
u
v
1
2
b
F
or
a
giv
en
in
teger
N
,
what
do
es
the
pair
of
v
alues
returned
b
y
the
function
f
corresp
ond
to
in
relation
to
the
situation
studied
in
this
exercise?
P
art
B
1
a
Using
recursiv
e
reasoning,
sho
w
that,
for
an
y
natu-
ral
n
um
b
er,
w
e
ha
v
e
:
v
n
−
u
n
>
0
b
Deduce
that
the
sequence
u
n
is
an
increasing
se-
quence
and
v
n
is
a
decreasing
sequence.
2
Justify
that
the
sequences
u
n
and
v
n
are
con
v
ergen
t.
P
art
C
1
Consider
the
sequence
w
n
defined,
for
an
y
natural
n
um-
b
er
n
,
b
y
:
w
n
=
v
n
−
u
n
a
Demonstrate
that
the
sequence
w
n
is
geometric.
b
Justify
that
the
limits
of
the
sequences
u
n
and
v
n
are
equal.
2
Consider
the
sequence
t
n
defined,
for
an
y
natural
n
um-
b
er
n
,
b
y
:
t
n
=
2
·
u
n
+3
·
v
n
a
Demonstrate
that
the
sequence
t
n
is
constan
t.
b
Deduce
the
expression
of
the
terms
of
the
sequence
u
n
as
a
function
of
n
.
c
Determine
the
limit
of
the
sequence
u
n
.
3.
A
little
further
:
homo
gr
aphic
se
quenc
es
E.2454
Consider
the
sequence
u
n
n
∈
N
de-
fined
b
y
:
u
0
=
14
;
u
n
+1
=
10
·
u
n
−
1
u
n
+
8
for
an
y
n
∈
N
.
W
e
admit
that
the
sequence
u
n
is
defined
on
N
and
that
the
terms
of
the
sequence
are
strictly
greater
than
1
.
1
Sho
wn
b
elo
w
is
the
represen
tativ
e
curv
e
of
the
function
f
whose
image
of
x
is
defined
b
y
the
relation
:
f
(
x
)
=
10
x
−
1
x
+
8
as
w
ell
as
the
represen
tation
of
the
first
bisector
of
the
plane
:
Sho
w
the
first
fiv
e
terms
of
the
sequence
on
the
x-axis.
2
Consider
the
sequence
v
n
n
∈
N
defined
b
y
the
relation
:
v
n
=
1
u
n
−
1
for
an
y
natural
n
um
b
er
n
.
a
Sho
w
that
for
an
y
n
∈
N
,
w
e
ha
v
e
:
v
n
+1
−
v
n
=
1
9
b
Giv
e
the
explicit
form
ula
defining
eac
h
term
of
the
se-
quence
v
n
as
a
function
of
rank
n
.
c
Deduce
the
explicit
form
ula
defining
eac
h
term
of
the
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5900
sacados/5900
chapExoCorrec/2454
sacados/2454
0
2
4
6
8
10
12
14
2
4
6
8
-4
-3
-2
-1
2
I
-1
2
J
O
C
f
sequence
u
n
as
a
function
of
rank
n
.
E.2414
Consider
the
sequence
u
n
n
∈
N
de-
fined
:
u
0
=
4
;
u
n
+1
=
−
u
n
+6
u
n
−
2
for
an
y
natural
n
um
b
er
n
.
1
Determine
the
first
three
terms
of
the
sequence
u
n
.
2
Let
v
n
n
∈
N
b
e
the
sequence
defined
b
y
the
relation
:
v
n
=
u
n
+
2
u
n
−
3
for
an
y
natural
n
um
b
er
n
.
a
Determine
the
first
three
terms
of
this
sequence.
b
Sho
w
that
:
v
n
+1
v
n
=
−
1
4
c
Deduce
the
nature
of
the
sequence
v
n
as
w
ell
as
the
explicit
form
ula
determining
the
term
of
rank
n
as
a
function
of
n
.
3
a
Determine
the
expression
of
the
term
u
n
as
a
func-
tion
of
the
term
v
n
.
b
Deduce
the
explicit
form
ula
defining
the
terms
of
u
n
as
a
function
of
n
.
E.5166
Consider
the
function
f
defined
on
−∞
;
8
b
y
the
relation
:
f
(
x
)
=
4
·
x
+
4
−
x
+
8
In
the
orthonormal
reference
frame
O
;
I
;
J
b
elo
w,
is
repre-
sen
ted
the
curv
e
C
f
represen
tativ
e
of
the
function
f
.
Consider
the
sequence
u
n
defined
b
y
the
relations
:
u
0
=
−
4
;
u
n
+1
=
f
u
n
for
all
n
∈
N
1
On
the
graph,
plot
the
first
fiv
e
terms
of
the
suite
u
n
on
the
x-axis.
2
By
calculation,
determine
the
exact
v
alue
of
the
first
fiv
e
terms
of
the
sequence
u
n
.
3
Consider
the
sequence
v
n
defined
b
y
the
relation
:
v
n
=
1
u
n
−
2
for
all
n
∈
N
a
Determine
the
first
fiv
e
terms
of
the
sequence
v
n
.
b
Mak
e
a
conjecture
ab
out
the
nature
of
the
sequence
v
n
and
the
v
alue
of
its
c
haracteristic
elemen
ts.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/2414
sacados/2414
chapExoCorrec/5166
sacados/5166
-4
-3
-2
-1
2
I
-1
2
J
O
C
f
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