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/ Limit of sequences
62 exercises (100% corrected)
a
Reminders: arithmetic and geometric sequences
(7 exercices)
Reminders: other
(3 exercices)
Limits of arithmetic and geometric sequences
(5 exercices)
Sum limits of terms of sequences
(8 exercices)
Other limits
(3 exercices)
With a spreadsheet
(3 exercices)
Arithmetic-geometric sequences
(4 exercices)
Jointly defined sequences
(5 exercices)
Other suites
(6 exercices)
Other sequences: linear recurrent sequences of order 2
(2 exercices)
Limits of explicitly defined sequences
(2 exercices)
Gendarme theorems
(4 exercices)
Lesson - Sequences
(4 exercices)
image PNG : /home/_math/_exercice/d6/6722/metapost/bouton.png
1.
R
eminders
:
arithmetic
and
ge
ometric
se
quenc
es
E.3393
Consider
the
arithmetic
sequence
u
n
n
∈
N
of
first
term
3
4
and
reason
1
2
.
1
Determine
the
v
alue
of
the
first
fiv
e
terms
of
this
se-
quence.
2
Giv
e
the
explicit
form
ula
of
u
n
giving
the
v
alue
of
a
term
as
a
function
of
its
rank.
3
Determine
the
v
alue
of
the
follo
wing
sequence
:
S
=
u
5
+
u
6
+
·
·
·
+
u
12
E.3394
Consider
the
geometric
sequence
u
n
n
∈
N
of
first
term
16
27
and
reason
3
2
.
1
Determine
the
v
alue
of
the
first
fiv
e
terms
of
this
se-
quence.
2
Giv
e
the
explicit
form
ula
of
u
n
giving
the
v
alue
of
a
term
as
a
function
of
its
rank.
3
Determine
the
v
alue
of
the
follo
wing
sequence
:
S
=
u
3
+
u
4
+
·
·
·
+
u
16
E.5012
1
Let
u
n
b
e
an
arithmetic
sequence
defined
for
n
∈
N
.
W
e
ha
v
e
the
v
alue
of
the
follo
wing
t
w
o
terms
:
u
4
=
3
;
u
7
=
15
a
Determine
the
c
haracteristic
elemen
ts
of
the
suite
u
n
.
b
Giv
e
the
recurrence
form
ula,
then
the
explicit
form
ula
for
the
sequence
u
n
2
Let
v
n
b
e
a
geometric
sequence
defined
for
n
∈
N
.
W
e
ha
v
e
the
v
alue
of
the
follo
wing
t
w
o
terms
:
v
2
=
2
;
v
5
=
54
a
Determine
the
c
haracteristic
elemen
ts
of
the
suite
v
n
.
b
Giv
e
the
recurrence
form
ula,
then
the
explicit
form
ula
for
the
sequence
v
n
E.6724
Consider
the
t
w
o
sequences
of
n
um
b
ers
b
elo
w,
whose
first
sev
en
terms
are
giv
en
:
a
3
;
5
;
7
;
10
;
12
;
14
;
16
b
6
;
3.5
;
1
;
-1.5
;
-4
;
-6.5
;
-9
F
or
eac
h
of
the
questions,
can
w
e
conjecture
that
the
sequence
is
an
arithmetic
sequence?
If
y
es,
giv
e
the
first
term
and
the
reason.
If
no,
justify
y
our
rejection
of
this
statemen
t.
E.6725
Consider
the
t
w
o
sequences
of
n
um
b
ers
b
elo
w
où
giv
en
the
first
six
terms
:
a
8
;
4
;
2
;
1
;
1
2
;
1
4
b
1
;
3
;
9
;
18
;
54
;
162
F
or
eac
h
of
the
questions,
can
w
e
conjecture
that
the
sequence
is
a
geometric
sequence?
If
so,
sp
ecify
the
first
term
and
the
reason.
If
not,
justify
y
our
rejection
of
the
conjecture.
E.3398
Iden
tifying
eac
h
of
the
sums
as
a
sum
of
the
terms
of
an
arithmetic
or
geometric
sequence,
determine
eac
h
of
their
v
alues
:
a
12
+
7
+
2
+
(
−
3)
+
·
·
·
+
(
−
28)
b
27
+
3
+
1
3
+
·
·
·
+
1
243
c
2
3
+
8
3
+
14
3
+
·
·
·
+
62
3
d
1
2
4
+
1
2
6
+
1
2
8
+
·
·
·
+
1
2
24
E.3807
Consider
the
sequence
p
n
n
∈
N
∗
whose
first
six
terms
v
erify
:
The
first
six
terms
are
all
p
ositiv
e
and
their
sum
is
1
.
The
first
six
terms
of
the
sequence
are
non-constan
t.
The
first
six
terms
of
p
n
are
consecutiv
e
terms
of
an
arithmetic
sequence.
The
n
um
b
ers
p
1
,
p
2
,
p
4
are
three
consecutiv
e
terms
of
a
geometric
sequence.
Determine
the
v
alue
of
the
first
six
terms
of
the
sequence
(
p
n
)
.
2.
R
eminders
:
other
E.3395
1
Consider
the
sequence
u
n
n
∈
N
defined
b
y
:
u
0
=
1
u
1
=
1
u
n
+2
=
u
n
+1
+
u
n
p
our
tout
n
∈
N
Determine
the
v
alue
of
the
first
eigh
t
terms
of
the
se-
quence
u
n
.
2
Consider
the
sequence
v
n
n
N
∗
defined
b
y
:
v
1
=
2
;
v
n
+1
=
1
v
n
+
n
for
all
n
∈
N
Determine
the
first
fiv
e
terms
of
the
sequence
v
n
.
E.3396
Determine
the
v
alue
of
eac
h
of
the
fol-
lo
wing
sums
:
a
7
i
=0
i
b
8
i
=3
i
2
−
i
c
7
i
=0
i
−
4
d
6
i
=1
1
i
e
4
i
=1
1
i
2
f
3
=0
i
=0
i
h
ttps:/
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hingmath.fr
chapExoCorrec/3393
sacados/3393
chapExoCorrec/3394
sacados/3394
chapExoCorrec/5012
sacados/5012
chapExoCorrec/6724
sacados/6724
chapExoCorrec/6725
sacados/6725
chapExoCorrec/3398
sacados/3398
chapExoCorrec/3807
sacados/3807
chapExoCorrec/3395
sacados/3395
chapExoCorrec/3396
sacados/3396
E.5042
Justify
that,
in
eac
h
question,
the
infor-
mation
b
elo
w
do
es
not
define
sequences
:
a
u
0
=
5
;
u
n
+1
=
2
·
u
n
−
3
for
all
n
∈
N
∗
b
u
0
=
1
;
u
1
=
4
;
u
n
+1
=
u
n
−
3
for
all
n
∈
N
c
u
0
=
3
;
u
n
=
2
·
u
n
−
1
−
2
for
all
n
∈
N
d
u
0
=
−
1
;
u
n
=
u
n
−
1
−
2
u
n
−
1
+
1
for
all
n
∈
N
∗
3.
Limits
of
arithmetic
and
ge
ometric
se
quenc
es
E.6726
1
Consider
the
sequence
u
n
arithmetic
with
first
term
−
4
and
reason
5
:
a
Complete
the
table
b
elo
w
:
n
0
1
2
3
4
5
6
7
u
n
−
4
1
6
11
16
21
b
What
conjecture
can
b
e
made
ab
out
the
v
alue
of
the
term
u
n
as
the
rank
n
b
ecomes
larger
and
larger?
Note
:
lim
n
↦→
+
∞
u
n
=
:
:
:
:
:
:
2
Consider
the
sequence
v
n
arithmetic
of
first
term
3
and
reason
−
1.2
:
a
Complete
the
table
b
elo
w
:
n
0
1
2
3
4
5
6
7
v
n
3
1.8
0.6
−
0.6
−
1.8
−
3
b
What
conjecture
can
b
e
made
ab
out
the
v
alue
of
the
term
v
n
as
the
rank
n
b
ecomes
larger
and
larger?
Note
:
lim
n
↦→
+
∞
v
n
=
:
:
:
:
:
:
E.6727
1
Consider
the
sequence
u
n
geometric
with
first
term
4
and
reason
2
:
a
Complete
the
table
b
elo
w
:
n
0
1
2
3
4
5
6
7
u
n
4
8
16
32
64
128
b
What
conjecture
can
b
e
made
ab
out
the
v
alue
of
the
term
u
n
as
the
rank
n
b
ecomes
larger
and
larger?
Note
:
lim
n
↦→
+
∞
u
n
=
:
:
:
:
:
:
2
Consider
the
sequence
v
n
geometric
with
first
term
81
and
reason
1
3
:
a
Complete
the
table
b
elo
w
:
n
0
1
2
3
4
5
6
7
v
n
81
27
9
3
1
1
3
b
What
conjecture
can
b
e
made
ab
out
the
v
alue
of
the
term
v
n
as
the
rank
n
b
ecomes
larger
and
larger?
Note
:
lim
n
↦→
+
∞
v
n
=
:
:
:
:
:
:
E.6728
Consider
the
sequence
u
n
geometric
with
first
term
2.8
and
reason
0.9
.
1
Using
the
calculator,
complete
the
table
b
elo
w
using
v
al-
ues
appro
ximated
to
the
h
undredth
:
n
0
1
2
3
4
5
6
7
u
n
2.8
2.52
2.268
2.041
2
Using
the
calculator,
what
conjecture
can
b
e
made
for
the
limit
of
the
terms
in
the
sequence
u
n
?
W
e
note
:
lim
n
↦→
+
∞
u
n
=
:
:
:
E.2557
1
Consider
the
sequence
u
n
n
∈
N
defined
explicitly
b
y
:
u
n
=
9
n
−
5
a
Determine
the
nature
of
the
sequence
u
n
,
sp
ecifying
its
c
haracteristics.
b
Determine
the
limit
of
the
sequence
u
n
.
2
Consider
the
sequence
v
n
n
∈
N
defined
explicitly
b
y
:
v
n
=
2
·
1
3
n
a
Determine
the
nature
of
the
sequence
v
n
,
sp
ecifying
its
c
haracteristics.
b
Determine
the
limit
of
the
sequence
v
n
.
E.6729
Giv
e,
if
p
ossible,
the
limits
of
the
follo
w-
ing
sequences
:
1
The
sequence
u
n
is
geometric
with
strictly
negativ
e
first
term
and
reason
2
.
2
The
sequence
v
n
is
geometric
with
strictly
p
ositiv
e
first
term
and
reason
−
3
.
3
The
suite
w
n
is
geometric
with
strictly
negativ
e
first
term
and
reason
−
0.2
.
4.
Sum
limits
of
terms
of
se
quenc
es
E.2559
1
Let
u
n
n
∈
N
b
e
the
arithmetic
sequence
of
first
term
2
and
reason
−
1
.
a
Determine
the
explicit
expression
of
the
terms
of
the
sequence
as
a
function
of
rank
n
.
b
Note
S
n
=
u
0
+
u
1
+
···
+
u
n
the
sum
of
the
first
(
n
+1)
terms
of
the
sequence.
Giv
e
the
expression
of
S
n
as
a
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5042
sacados/5042
chapExoCorrec/6726
sacados/6726
chapExoCorrec/6727
sacados/6727
chapExoCorrec/6728
sacados/6728
chapExoCorrec/2557
sacados/2557
chapExoCorrec/6729
sacados/6729
chapExoCorrec/2559
sacados/2559
function
of
n
.
c
Deduce
the
limit
:
lim
n
↦→
+
∞
S
n
.
2
Let
v
n
n
∈
N
b
e
the
geometric
sequence
of
first
term
5
and
reason
1
2
.
a
Determine
the
explicit
expression
of
the
terms
of
the
sequence
as
a
function
of
rank
n
.
b
Note
S
n
=
v
0
+
v
1
+
···
+
v
n
the
sum
of
the
first
(
n
+1)
terms
of
the
sequence.
Giv
e
the
expression
of
S
n
as
a
function
of
n
.
c
Deduce
the
limit
:
lim
n
↦→
+
∞
S
n
.
E.2588
Let
u
n
n
∈
N
b
e
a
geometric
sequence
of
first
term
2
and
reason
2
5
:
1
Determine
the
first
three
terms
of
this
sequence.
2
a
Determine
the
expression
of
the
sum
of
the
first
n
+1
terms
of
this
sequence
as
a
function
of
n
.
b
Deduce
the
v
alue
of
the
follo
wing
limit
:
lim
n
↦→
+
∞
u
0
+
u
1
+
·
·
·
+
u
n
E.2622
Consider
the
sequence
u
n
n
∈
N
geomet-
ric
with
first
term
4
×
5
2
and
reason
1
5
and
sum
S
n
defined
b
y
:
S
n
=
u
3
+
u
4
+
·
·
·
+
u
n
for
an
y
n
5
1
Determine
the
expression
of
S
n
as
a
function
of
n
.
2
Justify
that
the
sequence
S
n
con
v
erges
to
1.
E.2621
Consider
the
sequence
u
n
n
∈
N
geomet-
ric
of
first
term
1
and
reason
1
2
and
the
sequence
(
R
n
)
defined,
for
n
2
,
b
y
the
sum
:
R
n
=
u
n
+
u
n
+1
+
·
·
·
+
u
2
n
Determine
the
limit
of
the
sequence
(
R
n
)
.
E.3105
Determine
and
justify
the
limits
of
the
follo
wing
amoun
ts
:
a
S
=
1
2
+
1
2
2
+
1
2
3
+
·
·
·
+
1
2
n
b
S
=
−
15
4
+5
×
3
4
2
−
5
×
3
4
3
+5
×
3
4
4
+
···
+5
×
−
3
4
n
E.6174
A
runner
sets
himself
a
c
hallenge
:
he
w
an
ts
to
circumna
vigate
Europ
e.
On
the
first
da
y
,
he
co
v
ers
50
k
m
.
Through
fatigue,
from
da
y
to
da
y
,
his
daily
distance
tra
v
elled
is
reduced
b
y
1
%
.
W
e
note
u
n
the
length
co
v
ered
b
y
the
runner
on
the
n
-th
da
y
.
Assuming
that
the
runner
con
tin
ues
his
run
indefinitely
,
w
e
obtain
a
sequence
u
n
defined
for
an
y
non-zero
natural
n
um
b
er.
1
Determine
the
v
alue
of
the
first
four
terms
of
the
se-
quence
u
n
.
2
a
What
is
the
nature
of
the
sequence
u
n
?
Giv
e
the
c
haracteristic
elemen
ts
of
the
sequence
u
n
.
b
Express
the
term
u
n
as
a
function
of
rank
n
.
c
What
distance
will
the
runner
co
v
er
on
100
e
da
y?
W
e’ll
round
the
v
alue
to
the
ten
th
of
a
kilometer.
3
Note
S
the
sum
of
the
first
n
terms
of
the
sequence
u
n
:
S
n
=
u
1
+
u
2
+
·
·
·
+
u
n
a
Express
the
sum
S
n
as
a
function
of
rank
n
.
b
Complete
the
follo
wing
table,
rounding
v
alues
to
the
nearest
ten
th
of
a
kilometer
:
n
10
100
500
750
1000
S
n
c
What
conjecture
can
b
e
made
ab
out
the
limit
of
the
sum
S
n
when
the
v
alue
of
n
b
ecomes
larger
and
larger?
E.2560
A
glob
etrotter
has
b
et
to
co
v
er
5
000
k
m
on
fo
ot.
He
can,
fresh
and
ready
,
co
v
er
50
k
m
in
a
da
y
,
but
ev
ery
da
y
fatigue
builds
up
and
so
his
p
erformance
decreases
b
y
1
%
ev
ery
da
y
.
W
e’ll
note
d
n
the
distance
co
v
ered
during
the
n
-th
da
y
.
1
Calculate
the
distances
d
1
,
d
2
,
d
3
co
v
ered
during
the
first
three
da
ys.
2
What
precisely
is
the
nature
of
the
sequence?
Determine
the
v
alue
of
d
n
as
a
function
of
n
.
3
Note
L
n
the
distance
in
kilometers
tra
v
eled
after
n
da
ys.
L
n
=
d
1
+
d
2
+
·
·
·
+
d
n
a
Determine
the
expression
of
L
n
as
a
function
of
n
.
b
Deduce
the
limit
of
L
n
when
n
tends
to
w
ards
+
∞
.
Can
the
glob
etrotter
win?
c
Using
the
calculator,
determine
the
minim
um
n
um
b
er
of
da
ys
N
it
w
ould
tak
e
him
to
tra
v
el
4
999
k
m
E.5738
Consider
the
sequence
u
n
de-
fined
on
N
b
y
:
u
0
=
2
;
u
n
=
2
×
2
3
n
+
n
F
or
an
y
non-zero
natural
n
um
b
er
n
,
w
e
p
ose
:
S
n
=
n
k
=0
u
k
=
u
0
+
·
·
·
+
u
n
;
T
n
=
S
n
n
2
1
Express
the
term
S
n
in
terms
of
n
.
2
Determine
the
limit
of
the
sequence
T
n
.
5.
Other
limits
E.3397
Consider
the
function
f
defined
on
[
−
4
;
4]
whose
represen
tativ
e
curv
e
C
f
is
giv
en
b
elo
w
:
h
ttps:/
/c
hingmath.fr
chapExoCorrec/2588
sacados/2588
chapExoCorrec/2622
sacados/2622
chapExoCorrec/2621
sacados/2621
chapExoCorrec/3105
sacados/3105
chapExoCorrec/6174
sacados/6174
chapExoCorrec/2560
sacados/2560
extrait de Polynesie - 2006 - Obligatoire - 4 points
chapExoCorrec/5738
sacados/5738
chapExoCorrec/3397
sacados/3397
-4
-3
-2
-1
2
3
4
I
-4
-3
-2
-1
2
3
J
O
C
f
2
3
I
2
J
O
(
d
-3
-2
-1
2
3
4
5
6
I
2
3
4
5
6
J
O
C
f
Consider
the
sequence
u
n
n
∈
N
defined
b
y
the
relation
:
u
0
=
2
;
u
n
+1
=
f
u
n
1
Sho
w
that
the
term
u
1
is
equal
to
−
3
.
2
Justify
the
follo
wing
equalities
:
a
u
2
=
−
0.5
b
u
3
=
2.5
3
Complete
the
follo
wing
table
:
n
0
1
2
3
4
5
6
7
8
9
u
n
4
What
can
b
e
said
ab
out
the
limit
of
the
terms
of
the
sequence
u
n
?
E.3411
Consider
the
sequence
u
n
n
∈
N
defined
b
y
the
recurrence
relation
:
u
0
=
5
2
;
u
n
+1
=
−
1
2
·
u
n
+
3
2
for
an
y
n
∈
N
The
graph
b
elo
w
represen
ts,
in
a
reference
frame
O
;
I
;
J
orthonormal,
the
straigh
t
line
(
d
)
ha
ving
equation
:
y
=
−
1
2
·
x
+
3
2
1
Graphically
,
on
the
x-axis
represen
ted
the
first
fiv
e
terms
of
this
sequence
(c
onstructions
must
b
e
left)
.
2
a
Using
a
calculator,
complete
the
table
b
elo
w,
giving
v
alues
appro
ximated
to
the
nearest
ten
th
:
n
0
1
2
3
4
5
6
7
8
9
u
n
2.5
0.25
1.38
0.81
1.09
b
What
conjecture
can
b
e
made
ab
out
the
limit
of
the
sequence
u
n
?
E.3412
Consider
the
sequence
u
n
n
∈
N
defined
b
y
the
recurrence
relation
:
u
0
=
−
1
;
u
n
+1
=
f
u
n
for
all
n
∈
N
where
the
function
f
is
defined
b
y
:
f
(
x
)
=
x
2
+7
·
x
+14
2
·
x
+3
The
graph
b
elo
w
represen
ts,
in
an
orthonormal
co
ordinate
system
O
;
I
;
J
,
the
curv
e
C
f
represen
ting
the
function
f
.
1
Graphically
represen
t
the
first
fiv
e
terms
of
this
sequence
on
the
x-axis
(the
c
onstructions
must
b
e
left)
.
2
a
Using
a
calculator,
complete
the
table
b
elo
w
b
y
indi-
cating
the
v
alues
of
the
terms
rounded
to
t
w
o
decimal
places.
n
0
1
2
3
4
5
u
n
−
1
2
3
;
2
3
;
76
4
;
03
n
6
7
8
9
10
11
u
n
b
What
conjecture
can
b
e
made
ab
out
the
limit
of
the
terms
of
the
sequence
u
n
?
6.
With
a
spr
e
adshe
et
h
ttps:/
/c
hingmath.fr
-4
-3
-2
-1
2
3
4
I
-4
-3
-2
-1
2
3
J
O
C
f
chapExoCorrec/3411
sacados/3411
2
3
I
2
J
O
(
d
chapExoCorrec/3412
sacados/3412
-3
-2
-1
2
3
4
5
6
I
2
3
4
5
6
J
O
C
f
1
2
3
4
5
6
A
B
C
D
E
x
n
y
n
w
n
t
n
1
12
8,33
9,25
8,94
9,02
9,00
9,00
9,00
9,00
1
2
3
4
5
6
7
A
B
x
n
y
n
-3
4
-4,8
1,4
-4,68
1,76
-2,688
-4,216
0,3792
-4,9856
3,29472
-3,76096
E.6732
Use
the
calculator
to
complete
the
ta-
ble
of
v
alues
for
eac
h
of
the
follo
wing
sequences.
If
necessary
,
round
the
v
alues
to
the
nearest
h
undredth
:
1
Let
u
n
b
e
defined
b
y
the
relation
:
u
n
=
2
·
n
+
3
for
an
y
n
∈
N
n
0
1
2
3
4
5
6
7
u
n
2
Let
v
n
b
e
defined
b
y
the
relation
:
v
0
=
5
;
v
n
+1
=
0.75
·
v
n
+
3
for
an
y
n
∈
N
Let
w
n
b
e
defined
b
y
the
relation
:
w
n
=
v
n
−
12
for
an
y
n
∈
N
a
Complete
the
table
of
v
alues
:
n
0
1
2
3
4
5
6
7
v
n
w
n
b
V
erify
that
the
8
terms
of
the
sequence
w
n
allo
w
us
to
conjecture
that
the
sequence
w
n
is
geometric.
3
Let
t
n
b
e
defined
b
y
the
relationship
:
t
0
=
0
;
t
1
=
1
;
t
n
+1
=
t
n
+
t
n
−
1
for
all
n
∈
N
n
0
1
2
3
4
5
6
7
t
n
4
Let
a
n
and
b
n
b
e
defined
b
y
the
relations
:
a
0
=
2
b
0
=
−
1
;
a
n
+1
=
2
·
a
n
−
b
n
b
n
+1
=
a
n
+
3
·
b
n
for
an
y
n
∈
N
n
0
1
2
3
4
5
6
7
a
n
b
n
E.6731
W
e
define
the
t
w
o
sequences
u
and
v
b
y
:
u
0
=
1
v
0
=
12
;
u
n
+1
=
1
3
·
u
n
+
2
·
v
n
v
n
+1
=
1
4
·
u
n
+
3
·
v
n
for
an
y
n
∈
N
1
Using
a
spreadsheet,
generate
the
first
20
terms
of
these
t
w
o
sequences
to
obtain
a
table
sim-
ilar
to
the
one
sho
wn
opp
osite.
2
W
e
define
the
sequence
w
n
b
y
:
w
n
=
v
n
−
u
n
a
In
the
column
C
,
express
the
first
20
terms
of
the
sequence
w
.
b
What
can
b
e
done
to
demonstrate
that
the
sequence
w
follo
ws
a
geometric
progression
o
v
er
its
first
20
terms?
2
W
e
define
the
sequence
t
defined
b
y
:
t
n
=
3
·
u
n
+8
·
v
n
a
Express
the
first
20
terms
of
the
sequence
t
.
b
What
conjecture
can
b
e
made
ab
out
the
nature
of
the
sequence
t
?
E.6722
Consider
the
t
w
o
sequences
x
n
and
y
n
join
tly
defined
b
y
the
relations
:
x
0
=
−
3
y
0
=
4
et
x
n
+1
=
0.8
·
x
n
−
0.6
·
y
n
y
n
+1
=
0.6
·
x
n
+
0.8
·
y
n
for
all
n
∈
N
1
Using
a
spreadsheet,
generate
the
first
100
terms
of
the
sequence
in
a
table
similar
to
the
one
sho
wn
opp
o-
site.
2
Use
the
graphics
to
ol,
selecting
the
range
A2:B101
to
represen
t
in
the
plane
the
p
oin
ts
M
n
defined
b
y
M
n
(
x
n
;
y
n
)
.
The
represen
tation
mo
de
ˇ
XY
(dis-
p
ersion)
ı
will
b
e
c
hosen.
3
a
What
conjecture
can
b
e
made
ab
out
the
p
osition
of
the
sequence
of
p
oin
ts
M
n
in
the
plane?
b
V
erify
that
the
p
oin
t
M
0
v
erifies
this
conjecture.
c
Establish
the
relationship
:
O
M
n
+1
=
O
M
n
d
Complete
the
follo
wing
sen
tence
:
ˇ
If,
for
an
y
natural
n
um
b
er,
the
terms
of
a
se-
quence
v
erify
the
relation
u
n
+1
=
u
n
then
the
se-
quence
is
.
.
.
.
.
.
.
.
.
ı
7.
A
rithmetic-ge
ometric
se
quenc
es
E.6733
Consider
the
sequence
u
n
defined
b
y
the
relation
:
u
0
=
8
;
u
n
+1
=
0.95
·
u
n
+
0.5
for
an
y
n
∈
N
1
W
e
define
the
sequence
v
n
defined
b
y
the
relation
:
v
n
=
u
n
−
10
for
all
n
∈
N
a
Justify
that
the
sequence
v
n
is
a
geometric
sequence
of
reason
0.95
.
Sp
ecify
the
v
alue
of
its
first
term.
b
Express
the
terms
of
the
sequence
v
n
as
a
function
of
n
.
2
Deduce
an
expression
for
the
sequence
u
n
as
a
function
of
n
3
Determine
the
limit
of
the
terms
of
the
sequence
u
n
when
n
tends
to
+
∞
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6732
sacados/6732
chapExoCorrec/6731
sacados/6731
1
2
3
4
5
6
A
B
C
D
E
x
n
y
n
w
n
t
n
1
12
8,33
9,25
8,94
9,02
9,00
9,00
9,00
9,00
chapExoCorrec/6722
sacados/6722
1
2
3
4
5
6
7
A
B
x
n
y
n
-3
4
-4,8
1,4
-4,68
1,76
-2,688
-4,216
0,3792
-4,9856
3,29472
-3,76096
chapExoCorrec/6733
sacados/6733
E.6734
Consider
the
sequence
u
n
defined
b
y
the
relation
:
u
0
=
−
2
;
u
n
+1
=
2
·
u
n
+
0.5
for
an
y
n
∈
N
1
W
e
define
the
sequence
v
n
defined
b
y
the
relation
:
v
n
=
u
n
+
0.5
for
all
n
∈
N
a
Justify
that
the
sequence
v
n
is
a
geometric
sequence
whose
c
haracteristic
elemen
ts
will
b
e
sp
ecified
b
Express
the
terms
of
the
sequence
v
n
as
a
function
of
n
.
2
Deduce
an
expression
for
the
sequence
u
n
as
a
function
of
n
3
Determine
the
limit
of
the
terms
of
the
sequence
u
n
when
n
tends
to
+
∞
.
E.3399
W
e
wish
to
study
the
sequence
(
u
n
)
of
first
term
u
0
=
5
defined
b
y
the
follo
wing
recurrence
relation
:
u
n
+1
=
1
3
u
n
+
4
for
all
n
∈
N
W
e
define
the
sequence
v
n
b
y
:
v
n
=
u
n
−
6
for
all
n
∈
N
1
Sho
w
that
the
sequence
(
v
n
)
is
a
geometric
sequence
whose
first
term
and
reason
will
b
e
sp
ecified.
2
Express
v
n
as
a
function
of
rank
n
.
3
Deduce
the
expression
of
u
n
as
a
function
of
n
.
4
Deduce
the
limit
of
the
sequence
u
n
.
E.5740
Consider
the
sequence
u
n
)
de-
fined
on
N
∗
b
y
:
u
1
=
0
;
u
n
+1
=
0.2
·
u
n
+
0.04
for
all
n
∈
N
∗
W
e
admit
that
for
an
y
natural
n
um
b
er
:
u
n
<
0.05
W
e
define
the
sequence
v
n
on
N
∗
b
y
the
relation
:
v
n
=
u
n
−
0.05
1
Sho
w
that
the
sequence
v
n
is
a
geometric
sequence
whose
reason
will
b
e
sp
ecified.
2
Determine
the
expression
of
the
terms
of
the
sequence
v
n
as
a
function
of
n
.
Deduce
the
expression
of
the
terms
of
the
sequence
u
n
as
a
function
of
n
.
3
Deduce
the
limit
of
the
sequence
u
n
.
8.
Jointly
define
d
se
quenc
es
E.6739
Consider
the
t
w
o
sequences
u
n
and
v
n
join
tly
defined
b
y
the
follo
wing
relationships
:
u
0
=
5
v
0
=
4
;
u
n
+1
=
2
·
u
n
−
v
n
v
n
+1
=
−
3
·
u
n
+
4
·
v
n
for
an
y
n
∈
N
1
Consider
the
sequence
w
n
defined
b
y
the
relation
:
w
n
=
v
n
−
u
n
for
all
n
∈
N
a
Establish
that
the
sequence
w
n
is
a
geometric
se-
quence
of
reason
5
.
b
Deduce
that,
for
an
y
natural
n
um
b
er
n
,
the
term
of
rank
n
is
expressed
as
:
w
n
=
−
5
n
2
Consider
the
sequence
t
n
defined
b
y
:
t
n
=
3
·
u
n
+
v
n
a
Sho
w
that
:
t
0
=
19
b
Establish
that
for
an
y
natural
n
um
b
er,
w
e
ha
v
e
equal-
it
y
:
t
n
+1
=
t
n
for
an
y
n
∈
N
W
e
will
admit
that,
for
an
y
n
∈
N
,
w
e
ha
v
e
:
t
n
=
19
3
Deduce
an
expression
for
the
terms
of
the
sequences
u
n
and
v
n
as
a
function
of
n
.
4
Deduce
the
limits
of
the
sequences
u
n
and
v
n
.
E.6738
Consider
the
t
w
o
sequences
u
n
and
v
n
join
tly
defined
b
y
the
follo
wing
relations
:
u
0
=
3
v
0
=
−
1
;
u
n
+1
=
3
·
u
n
−
2
·
v
n
v
n
+1
=
−
u
n
+
2
·
v
n
1
Consider
the
sequence
w
n
defined
b
y
the
relation
:
w
n
=
v
n
−
u
n
for
all
n
∈
N
a
Establish
that
the
sequence
w
n
is
a
geometric
se-
quence
of
reason
4
.
b
Deduce
that,
for
an
y
natural
n
um
b
er
n
,
the
term
of
rank
n
is
expressed
as
:
w
n
=
−
4
×
4
n
2
Consider
the
sequence
t
n
defined
b
y
:
t
n
=
4
·
u
n
+
8
·
v
n
for
an
y
n
∈
N
a
Giv
e
the
v
alue
of
the
term
t
0
.
b
Establish
that
for
an
y
natural
n
um
b
er,
w
e
ha
v
e
the
equalit
y
:
t
n
+1
=
t
n
W
e’ll
admit
that
the
sequence
t
n
is
constan
t.
3
Deduce
an
expression
for
the
terms
of
the
sequences
u
n
and
v
n
as
a
function
of
n
.
4
Deduce
the
limits
of
the
sequences
u
n
and
v
n
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6734
sacados/6734
chapExoCorrec/3399
sacados/3399
chapExoCorrec/5740
sacados/5740
chapExoCorrec/6739
sacados/6739
chapExoCorrec/6738
sacados/6738
E.3414
Consider
the
t
w
o
sequences
p
n
n
∈
N
and
q
n
n
∈
N
whose
first
terms
are
:
p
0
=
5
;
q
0
=
2
They
are
defined
b
y
the
follo
wing
recurrence
relations
:
p
n
+1
=
0.5
·
p
n
+
0.4
·
q
n
;
q
n
+1
=
0.4
·
p
n
+
0.5
·
q
n
W
e
define
the
sequences
u
n
n
∈
N
and
v
n
n
∈
N
b
y
the
follo
w-
ing
relations
:
u
n
=
q
n
+
p
n
;
v
n
=
p
n
−
q
n
1
Determine
the
exact
v
alue
of
the
first
three
terms
of
eac
h
of
the
sequences
u
n
and
v
n
.
2
a
Sho
w
that
the
sequence
u
n
is
a
geometric
sequence
of
reason
0.9
.
b
Deduce
the
limit
of
the
sequence
u
n
.
3
Determine
the
limit
of
the
sequence
v
n
.
4
a
Noting
that
u
n
+
v
n
=
2
·
p
n
,
deduce
that
the
sequence
p
n
is
con
v
ergen
t
;
sp
ecify
its
limit.
b
Establish
that
the
sequence
q
n
is
con
v
ergen
t
;
deter-
mine
its
limit.
E.6759
Consider
the
sequences
a
n
and
b
n
defined
b
y
:
a
0
=
6
b
0
=
1
a
n
+1
=
0.2
·
a
n
+
1.3
·
b
n
b
n
+1
=
−
0.8
·
a
n
−
0.2
·
b
n
for
an
y
n
∈
N
1
Express
the
terms
a
n
+2
and
b
n
+2
in
terms
of
a
n
and
b
n
.
2
What
can
b
e
said
ab
out
the
terms
of
the
sequence
a
2
n
?
E.6758
Consider
the
sequences
a
n
and
b
n
defined
b
y
:
a
0
=
2
b
0
=
1
;
a
n
+1
=
0.8
·
a
n
+
0.9
·
b
n
b
n
+1
=
0.4
·
a
n
−
0.8
·
b
n
∀
n
∈
N
1
Establish
the
equalities
:
a
n
+2
=
a
n
;
b
n
+2
=
b
n
2
What
can
w
e
sa
y
ab
out
the
limits
of
the
sequences
a
n
and
b
n
?
9.
Other
suites
E.6765
Let
u
n
b
e
the
sequence
defined
b
y
its
first
term
u
0
=
5
and,
for
an
y
natural
n
um
b
er
n
b
y
:
u
n
+1
=
0.5
·
u
n
+
0.5
·
n
−
1.5
1
Let
v
n
b
e
the
sequence
defined
for
an
y
natural
n
um
b
er
n
b
y
:
v
n
=
0.1
·
u
n
−
0.1
·
n
+
0.5
Sho
w
that
the
sequence
v
n
is
geometric
of
reason
0.5
and
then
express
v
n
as
a
function
of
n
.
2
Deduce
that,
for
an
y
natural
n
um
b
er
n
:
u
n
=
10
×
0.5
n
+
n
−
5
3
Then
determine
the
limit
of
the
sequence
u
n
E.6799
Consider
the
sequence
u
n
defined
b
y
:
u
0
=
6
;
u
n
+1
=
2
·
u
n
−
3
·
n
+
5
for
all
n
∈
N
1
Giv
e
the
first
three
terms
of
the
sequence
u
n
.
2
Consider
the
sequence
v
n
defined
b
y
:
v
n
=
u
n
−
3
·
n
+
2
a
Demonstrate
that
the
sequence
v
n
is
a
geometric
se-
quence
of
reason
2
.
b
Deduce
an
expression
for
the
terms
of
the
sequence
v
n
as
a
function
of
their
rank.
3
a
Justify
that
for
an
y
natural
n
um
b
er
n
,
w
e
ha
v
e
:
u
n
=
8
×
2
n
+
3
·
n
−
2
b
Deduce
the
limit
of
suite
u
n
.
E.5737
Let
the
n
umerical
sequence
u
n
b
e
defined
on
N
b
y
:
u
0
=
2
;
u
n
+1
=
2
3
·
u
n
+
1
3
·
n
+
1
for
all
n
∈
N
W
e
denote
b
y
v
n
the
sequence
defined
on
N
b
y
:
v
n
=
u
n
−
n
1
Demonstrate
that
the
sequence
v
n
is
a
geometric
se-
quence
of
reason
2
3
.
2
Deduce
that
for
an
y
natural
n
um
b
er
n
:
u
n
=
2
·
2
3
n
+
n
3
Determine
the
limit
of
the
sequence
u
n
.
E.2620
Consider
the
sequence
u
n
n
∈
N
whose
first
term
is
0
defined
b
y
the
follo
wing
recurrence
relation
:
u
n
+1
=
2
5
·
u
n
−
3
·
n
−
8
p
our
tout
n
∈
N
1
Determine
the
first
three
terms
of
this
sequence.
2
Consider
the
sequence
v
n
n
∈
N
defined
b
y
the
relation
:
v
n
=
u
n
+
5
n
+
5
for
all
n
∈
N
a
Sho
w
that
the
sequence
v
n
is
a
geometric
sequence
whose
reason
and
first
term
will
b
e
sp
ecified.
b
Deduce
the
explicit
form
ula
for
the
sequence
u
n
of
a
term
as
a
function
of
n
its
rank.
3
Deduce
the
limit
of
the
sequence
u
n
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3414
sacados/3414
chapExoCorrec/6759
sacados/6759
chapExoCorrec/6758
sacados/6758
chapExoCorrec/6765
sacados/6765
chapExoCorrec/6799
sacados/6799
chapExoCorrec/5737
sacados/5737
chapExoCorrec/2620
sacados/2620
E.3285
W
e
define
the
sequence
u
n
is
defined
b
y
:
u
0
=
1
;
u
n
+1
=
1
2
u
n
+
n
−
1
for
all
n
∈
N
W
e
define
the
sequence
(
v
n
)
b
y
:
v
n
=
4
u
n
−
8
n
+24
1
By
successiv
e
transformation,
establish
the
follo
wing
equalit
y
:
v
n
+1
=
1
2
·
v
n
.
2
Deduce
an
expression
for
the
terms
of
the
sequence
u
n
as
a
function
of
n
.
E.3518
Consider
the
n
umerical
sequence
u
defined
b
y
:
u
0
=
1
;
u
n
+1
=
1
3
·
u
n
+
n
−
1
for
an
y
natural
n
um
b
er
n
.
Let
v
b
e
the
sequence
defined
for
an
y
in
teger
n
taturel,
b
y
:
v
n
=
4
·
u
n
−
6
n
+
15
1
Sho
w
that
v
is
a
geometric
sequence.
2
Calculate
v
0
,
then
calculate
v
n
as
a
function
of
n
.
Deduce
that
for
an
y
natural
n
um
b
er
n
:
u
n
=
19
4
×
1
3
n
+
6
n
−
15
4
3
Sho
w
that
the
sequence
u
can
b
e
written
as
:
u
=
t
+
w
où
t
is
a
geometric
sequence
w
a
sequence
arithmétique
4
Calculate
the
v
alue
of
the
t
w
o
sums
:
T
n
=
t
0
+
t
1
+
·
·
·
+
t
n
;
W
n
=
w
0
+
w
1
+
·
·
·
+
w
n
.
Deduct
:
U
n
=
u
0
+
u
1
+
···
+
u
n
.
10.
Other
se
quenc
es
:
line
ar
r
e
curr
ent
se
quenc
es
of
or
der
2
E.3515
Consider
the
sequence
u
n
n
∈
N
defined
b
y
:
u
0
=
0
u
1
=
1
;
u
n
+1
=
7
·
u
n
+
8
·
u
n
−
1
p
our
tout
n
∈
N
1
Determine
the
v
alue
of
the
first
fiv
e
terms
of
the
sequence
u
n
.
2
Sho
w
that
the
sequence
s
n
n
∈
N
defined
b
y
:
s
n
=
u
n
+1
+
u
n
p
our
tout
n
∈
N
is
a
geometric
sequence
whose
reason
will
b
e
sp
ecified.
Deduce
s
n
as
a
function
of
n
.
3
a
W
e
p
ose
v
n
=
(
−
1)
n
·
u
n
and
consider
the
sequence
t
n
n
∈
N
defined
b
y
:
t
n
=
v
n
+1
−
v
n
for
all
n
∈
N
Express
t
n
as
a
function
of
s
n
.
b
What
is
the
nature
of
the
sequence
t
n
.
4
a
Express
v
n
,
then
u
n
,
as
a
function
of
n
(we
c
an
c
al-
culate,
in
two
ways,
the
sum
t
0
+
···
+
t
n
−
1
)
.
b
Determine
:
lim
n
↦→
+
∞
u
n
8
n
.
E.3421
Consider
the
set
(
E
)
of
the
se-
quences
x
n
defined
on
N
and
v
erifying
the
follo
wing
rela-
tion
:
for
an
y
non-zero
natural
n
um
b
er
n
,
x
n
+1
−
x
n
=
0.24
·
x
n
−
1
1
Consider
a
non-zero
real
–
and
define
on
N
the
sequence
t
n
b
y
t
n
=
–
n
.
Sho
w
that
the
sequence
t
n
b
elongs
to
the
set
(
E
)
if,
and
only
if,
–
is
a
solution
of
the
equation
:
–
2
−
–
−
0.24
=
0
Deduce
the
sequences
t
n
b
elonging
to
the
set
(
E
)
.
Let
(
E
)
b
e
the
set
of
sequences
u
n
defined
on
N
b
y
a
rela-
tion
of
the
form
:
u
n
=
¸
·
(1.2)
n
+
˛
·
(
−
0.2)
n
¸
and
˛
are
real
n
um
b
ers.
2
Consider
a
sequence
u
n
of
the
set
(
E
)
.
Determine
the
v
alues
of
¸
and
˛
suc
h
that
:
u
0
=
6
and
u
1
=
6.6
Deduce
that,
for
an
y
natural
n
um
b
er
n
:
u
n
=
39
7
·
(1.2)
n
+
3
7
·
(
−
0.2)
n
.
3
Determine
:
lim
n
↦→
+
∞
u
n
.
11.
Limits
of
explicitly
define
d
se
quenc
es
E.2558
Determine
the
limits
of
the
sequences
u
n
defined
b
elo
w
:
a
n
3
×
5
n
b
n
−
2
7
n
c
1
3
n
−
3
2
n
d
8
n
−
3
n
e
5
n
−
2
n
3
n
+
2
n
f
31
7
n
·
2
8
n
E.3104
In
eac
h
case,
determine
the
limit
of
the
sequence
u
n
whose
general
term
as
a
function
of
n
is
giv
en
b
elo
w
a
n
5
×
3
n
2
3
b
5
n
−
8
n
c
2
3
n
·
4
3
n
+2
d
3
n
−
2
n
5
n
−
4
n
12.
Gendarme
the
or
ems
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3285
sacados/3285
chapExoCorrec/3518
sacados/3518
chapExoCorrec/3515
sacados/3515
chapExoCorrec/3421
sacados/3421
chapExoCorrec/2558
sacados/2558
chapExoCorrec/3104
sacados/3104
E.2554
Consider
the
sequence
u
n
n
∈
N
de-
fined,
for
an
y
natural
n
um
b
er
n
,
b
y
the
explicit
relation
:
u
n
=
3
n
+(
−
1)
n
2
n
−
1
1
Determine
the
first
three
terms
of
the
sequence
u
n
.
2
Establish
the
follo
wing
frame
:
F
or
an
y
n
∈
N
∗
:
3
n
−
1
2
n
−
1
u
n
3
n
+1
2
n
−
1
3
Deduce
the
con
v
ergence
v
alue
of
u
n
.
E.2567
Let
u
n
n
∈
N
b
e
defined
b
y
the
explicit
form
ula
of
its
rank
term
n
:
u
n
=
2
·
n
+
1
−
2
·
n
1
Sho
w
the
follo
wing
equalit
y
for
an
y
n
∈
N
:
u
n
=
1
2
·
n
+
1
+
2
·
n
2
F
or
an
y
non-zero
natural
n
um
b
er
n
,
establish
the
follo
w-
ing
frame
:
0
<
u
n
<
1
2
·
√
2
·
n
3
Deduce
the
limit
of
the
sequence
u
n
.
E.3441
Consider
the
sequence
u
n
n
∈
N
∗
whose
rank
term
n
is
defined
b
y
the
relation
:
u
n
=
1
n
+
1
+
1
n
+
2
+
·
·
·
+
1
n
+
n
1
Establish
the
follo
wing
frame
for
an
y
non-zero
natural
n
um
b
er
n
:
n
n
+
n
u
n
n
n
+
1
2
Deduce
the
con
v
ergence
of
the
sequence
u
n
;
sp
ecify
the
v
alue
of
the
limit.
E.2587
Consider
the
sequence
u
n
n
∈
N
∗
is
de-
fined
explicitly
b
y
the
form
ula
:
u
n
=
3
4
n
2
+
1
+
3
4
n
2
+
2
+
·
·
·
+
3
4
n
2
+
n
1
Compare
the
follo
wing
t
w
o
terms
:
3
4
n
2
+
1
and
3
4
n
2
+
n
2
Establish
that
the
sequence
u
n
con
v
erges
to
3
2
.
13.
L
esson
-
Se
quenc
es
E.5741
Let
‘
b
e
a
real
n
um
b
er.
Consider
a
se-
quence
u
n
that
con
v
erges
to
‘
then
the
real
‘
to
whic
h
the
sequence
u
n
con
v
erges
is
unique.
E.3281
Prerequisite
:
definition
of
a
se-
quence
tending
to
+
∞
.
ˇA
sequence
tends
to
+
∞
if,
for
an
y
real
A
,
all
the
terms
of
the
sequence
are,
from
a
certain
rank,
greater
than
A
ı
Pro
v
e
the
follo
wing
theorem
:
a
non-ma
jor
increasing
se-
quence
tends
to
+
∞
E.5496
Consider
the
t
w
o
sequences
u
n
and
v
n
v
erifying
:
There
exists
n
0
suc
h
that
from
n
0
,
the
terms
of
the
t
w
o
sequences
:
n
n
0
=
⇒
v
n
u
n
lim
n
↦→
+
∞
u
n
=
+
∞
Establish
that
:
lim
n
↦→
+
∞
v
n
=
+
∞
E.5497
Consider
a
gro
wing
sequence
u
n
.
If
the
sequence
u
n
con
v
erges
to
a
finite
limit
‘
then
all
terms
of
the
sequence
u
n
are
less
than
or
equal
to
‘
.
14.
Unclassifie
d
financial
ye
ars
E.5570
Consider
the
n
umerical
sequence
v
n
defined
for
an
y
natural
n
um
b
er
n
b
y
:
v
0
=
1
;
v
n
+1
=
9
6
−
v
n
for
an
y
in
teger
n
∈
N
Consider
the
sequence
w
n
defined
b
y
:
w
n
=
1
v
n
−
3
for
an
y
natural
n
um
b
er
n
.
1
Demonstrate
that
w
n
is
an
arithmetic
sequence
of
rea-
son
−
1
3
and
first
term
−
1
2
2
a
Determine
the
expression
of
the
terms
of
the
se-
quence
w
n
as
a
function
of
rank
n
.
b
Deduce
the
expression
of
the
terms
of
the
sequence
v
n
as
a
function
of
rank
n
.
(we
wil
l
not
simplify
the
expr
ession
of
v
n
)
.
3
Determine
the
limit
of
the
sequence
v
n
.
E.4012
Consider
the
sequence
of
real
n
um-
b
ers
u
n
defined
on
N
b
y
:
u
0
=
−
1
;
u
1
=
1
2
;
u
n
+2
=
u
n
+1
−
1
4
·
u
n
for
all
n
∈
N
1
Calculate
u
2
and
deduce
that
the
sequence
u
n
is
nei-
ther
arithmetic
nor
geometric.
2
W
e
define
the
sequence
v
n
b
y
p
osing
:
v
n
=
u
n
+1
−
1
2
·
u
n
for
all
n
∈
N
a
Calculate
v
0
.
b
Establish
that
v
n
is
geometric
of
reason
1
2
.
c
Express
v
n
in
terms
of
n
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/2554
sacados/2554
chapExoCorrec/2567
sacados/2567
chapExoCorrec/3441
sacados/3441
chapExoCorrec/2587
sacados/2587
chapExoCorrec/5741
sacados/5741
chapExoCorrec/3281
sacados/3281
chapExoCorrec/5496
sacados/5496
chapExoCorrec/5497
sacados/5497
chapExoCorrec/5570
sacados/5570
Extrait Liban 2013
chapExoCorrec/4012
sacados/4012
Extrait Antilles-Guyane Septembre 2010
E.4225
P
eter
and
Claude
pla
y
tennis.
Both
pla
y
ers
ha
v
e
the
same
c
hance
of
winning
the
first
game.
Thereafter,
when
P
eter
wins
a
game,
the
probabilit
y
that
he
wins
the
next
is
0.7
.
And
if
he
loses
one
game,
the
probabilit
y
that
he
will
lose
the
next
is
0.8
.
Throughout
the
exercise,
n
is
a
non-zero
natural
n
um
b
er.
Consider
the
ev
en
ts
:
G
n
:
ˇ
Pierre
wins
the
n
-game.
ı
P
n
:
ˇ
Pierre
loses
the
n
-th
game.
ı
W
e
p
ose
:
p
n
=
P
(
G
n
)
;
q
n
=
P
(
P
n
)
Finding
a
recurrence
relation
:
1
Determine
p
1
then
the
conditional
probabilities
:
P
G
1
(
G
2
)
;
P
P
1
(
G
2
)
.
2
Justify
the
equalit
y
:
p
n
+
q
n
=
1
.
3
Demonstrate
that
for
an
y
non-zero
natural
n
um
b
er
n
:
p
n
+1
=
0.5
·
p
n
+
0.2
.
E.2555
Determine
the
limits
of
the
sequences
u
n
n
∈
N
b
elo
w
defined
explicitly
:
a
u
n
=
2
n
2
−
3
n
+
1
n
+
1
b
u
n
=
n
−
3
n
2
+
1
c
u
n
=
2
n
+
1
n
+
1
d
u
n
=
1
+
n
−
2
n
2
+
3
n
3
E.6176
Determine
the
v
alue
of
the
follo
wing
limits
:
a
lim
n
↦→
+
∞
n
2
+
2
b
lim
n
↦→
+
∞
2
+
1
n
c
lim
n
↦→
+
∞
n
3
+
n
+
2
d
lim
n
↦→
+
∞
n
−
1
n
e
lim
n
↦→
+
∞
n
·
n
−
10
f
lim
n
↦→
+
∞
1
+
1
n
n
+
1
E.6177
Determine
the
v
alue
of
the
follo
wing
limits
:
a
lim
n
↦→
+
∞
n
2
−
n
b
lim
n
↦→
+
∞
n
2
+
n
+
1
n
c
lim
n
↦→
+
∞
1
n
2
+
1
n
d
lim
n
↦→
+
∞
n
n
2
+
n
+
1
e
lim
n
↦→
+
∞
2
n
2
+
n
+
1
n
−
2
f
lim
n
↦→
+
∞
n
2
−
1
n
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4225
sacados/4225
Extrait d'Asie Juin 2006
chapExoCorrec/2555
sacados/2555
chapExoCorrec/6176
sacados/6176
chapExoCorrec/6177
sacados/6177
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