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3
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1
Grade 12
/ Annals on sequences
22 exercises (including 21 corrected)
a
Suite studies
(7 exercices)
Suites, variations and limits
(8 exercices)
Suites and logarithms
(4 exercices)
Suites and integrations
(2 exercices)
1
2
3
4
5
6
7
8
A
B
C
n
0
1
2
3
4
u
2
4
9
24
63
v
7
14
28
56
Fonction
f(n)
u
0
Pour
i
allant
de
1
à
n
u
·
Fin
pour
Renvoyer
u
Fonction
f(n)
u
0
Pour
i
allant
de
0
à
n
−
1
u
·
Fin
pour
Renvoyer
u
Algorithme
1
Algorithme
2
0
1
2
3
4
5
6
7
8
9
10
11
12
20
40
60
80
100
120
140
160
0
0
1
2
2
6
3
12
4
20
5
30
6
42
7
56
8
72
9
90
10
110
11
132
12
156
1.
Suite
studies
E.6762
Let
u
b
e
the
sequence
defined
b
y
u
0
=
2
and,
for
an
y
natural
n
um
b
er
n
,
b
y
:
u
n
+1
=
2
·
u
n
+
2
·
n
2
−
n
Consider
also
the
sequence
v
defined,
for
an
y
natural
n
um
b
er
n
,
b
y
:
v
n
=
u
n
+
2
·
n
2
+
3
·
n
+
5
1
Here
is
an
extract
from
a
spreadsheet
:
What
form
ulas
ha
v
e
b
een
written
in
the
cells
C2
and
B3
and
copied
do
wn
to
displa
y
the
terms
of
the
sequences
u
and
v
?
2
Determine,
with
justification,
an
expression
for
v
n
and
u
n
as
a
function
of
n
only
.
E.6256
Consider
the
sequence
u
n
de-
fined
b
y
:
u
0
=
0
;
u
n
+1
=
u
n
+
2
·
n
+
2
for
an
y
n
∈
N
1
Calculate
u
1
and
u
2
.
2
Consider
the
t
w
o
functions
b
elo
w,
tak
en
from
algorithms,
taking
as
argumen
t
an
in
teger
n
:
Whic
h
of
these
t
w
o
functions
returns
the
v
alue
of
the
term
u
n
of
rank
n
,
the
v
alue
supplied
as
an
argumen
t
when
the
function
is
called?
3
Using
the
algorithm,
w
e
obtained
the
table
and
scatter-
plot
b
elo
w
où
n
figure
on
the
x-axis
and
u
n
on
the
y-axis.
a
What
conjecture
can
b
e
made
ab
out
the
direction
of
v
ariation
of
the
sequence
u
n
?
Demonstrate
this
conjecture.
b
The
parab
olic
shap
e
of
the
p
oin
t
cloud
leads
us
to
conjecture
the
existence
of
three
real
n
um
b
ers
a
,
b
and
c
suc
h
that,
for
an
y
natural
n
um
b
er
n
:
u
n
=
a
·
n
2
+
b
·
n
+
c
.
In
the
con
text
of
this
conjecture,
find
the
v
alues
of
a
,
b
and
c
using
the
information
pro
vided.
4
F
or
an
y
natural
n
um
b
er
n
,
the
sequence
v
n
is
defined
b
y
:
v
n
=
u
n
+1
−
u
n
a
Express
v
n
in
terms
of
the
natural
n
um
b
er
n
.
What
is
the
nature
of
the
sequence
v
n
?
b
W
e
define,
for
an
y
natural
n
um
b
er
n
:
S
n
=
n
k
=0
v
k
=
v
0
+
v
1
+
·
·
·
+
v
n
Sho
w
that,
for
an
y
natural
n
um
b
er
n
:
S
n
=
(
n
+
1)(
n
+
2)
c
Demonstrate
that,
for
an
y
natural
n
um
b
er
n
:
S
n
=
u
n
+1
−
u
0
,
then
express
u
n
as
a
function
of
n
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6762
sacados/6762
1
2
3
4
5
6
7
8
A
B
C
n
0
1
2
3
4
u
2
4
9
24
63
v
7
14
28
56
chapExoCorrec/6256
sacados/6256
Fonction
f(n)
u
0
Pour
i
allant
de
1
à
n
u
·
Fin
pour
Renvoyer
u
Fonction
f(n)
u
0
Pour
i
allant
de
0
à
n
−
1
u
·
Fin
pour
Renvoyer
u
Algorithme
1
Algorithme
2
0
1
2
3
4
5
6
7
8
9
10
11
12
20
40
60
80
100
120
140
160
0
0
1
2
2
6
3
12
4
20
5
30
6
42
7
56
8
72
9
90
10
110
11
132
12
156
E.5079
P
art
A
Consider
the
function
f
,
tak
en
from
an
algorithm,
and
whose
argumen
t
is
a
natural
n
um
b
er.
Function
f(N)
U
←
0
For
k
ranging
from
0
to
N
−
1
U
←
3
·
U
−
2
·
k+3
End
To
Return
U
What
is
the
v
alue
returned
b
y
this
function
when
called
with
the
v
alue
3
for
the
argumen
t
N
?
P
art
B
y
Consider
the
sequence
u
n
defined
b
y
u
0
=
0
and,
for
an
y
in
teger
nature
n
:
u
n
+1
=
3
·
u
n
−
2
·
n
+
3
1
Calculate
u
1
and
u
2
.
2
a
Demonstrate
b
y
recurrence
that,
for
an
y
natural
n
um
b
er
n
:
u
n
n
b
Deduce
the
limit
of
the
sequence
u
n
.
3
Demonstrate
that
the
sequence
u
n
is
increasing.
4
Let
the
sequence
v
n
b
e
defined,
for
an
y
natural
n
um
b
er
n
,
b
y
v
n
=
u
n
−
n
+
1
a
Demonstrate
that
the
sequence
v
n
is
a
geometric
se-
quence.
b
Deduce
that,
for
an
y
natural
n
um
b
er
n
:
u
n
=
3
n
+
n
−
1
5
Let
p
b
e
a
non-zero
natural
n
um
b
er.
a
Wh
y
can
w
e
sa
y
that
there
exists
at
least
one
in
teger
n
0
suc
h
that,
for
an
y
n
n
0
,
u
n
10
p
?
W
e
are
no
w
in
terested
in
the
smallest
in
teger
n
0
.
b
Justify
that
:
n
0
3
·
p
.
c
Using
the
calculator,
determine
this
in
teger
n
0
for
the
v
alue
p
=
3
.
d
Prop
ose
a
function
whic
h,
for
a
v
alue
p
passed
as
an
argumen
t,
returns
the
v
alue
of
the
smallest
in
teger
n
0
suc
h
that,
for
an
y
n
n
0
,
w
e
ha
v
e
u
n
10
p
.
E.5019
1
Consider
the
function
f
,
giv
en
b
elo
w
and
deriv
ed
from
an
algorithm,
taking
the
argumen
ts
a
,
b
,
N
of
non-zero
in
teger
v
alues
:
Function
f(a,b,N)
u
←
a
v
←
b
n
←
0
As
long
as
n
<
N
n
←
n+
1
u
←
a
+
b
2
v
←
a
2
+
b
2
2
a
←
u
b
←
v
.
End
As
long
as
Renvoyer
(
u
;
v
)
Repro
duce
and
complete
the
follo
wing
table,
sho
wing
the
v
alues
tak
en
b
y
the
function
v
ariables
f
during
step-b
y-
step
execution
and
when
the
the
call
is
made
with
the
v
alues
a
=
4
,
b
=
9
and
N
=
2
.
Successiv
e
v
alues
of
u
and
v
will
b
e
rounded
to
the
thousandth.
n
a
b
u
v
0
4
9
1
2
In
the
follo
wing,
a
and
b
are
t
w
o
real
n
um
b
ers
suc
h
that
0
<
a
<
b
.
Consider
the
sequences
u
n
and
v
n
defined
b
y
:
u
0
=
a
,
v
0
=
b
and,
for
an
y
natural
n
um
b
er
n
:
u
n
+1
=
u
n
+
v
n
2
;
v
n
+1
=
u
2
n
+
v
2
n
2
2
a
Demonstrate
b
y
recurrence
that,
for
an
y
natural
n
um
b
er
n
,
w
e
ha
v
e
:
u
n
>
0
;
v
n
>
0
b
Demonstrate
that,
for
an
y
natural
n
um
b
er
n
:
v
2
n
+1
−
u
2
n
+1
=
u
n
−
v
n
2
2
Deduce
that,
for
an
y
natural
n
um
b
er
n
,
w
e
ha
v
e
:
u
n
v
n
.
3
a
Demonstrate
that
the
sequence
u
n
is
increasing.
b
Compare
v
2
n
+1
and
v
2
n
.
Deduce
the
direction
of
v
aria-
tion
of
the
sequence
v
n
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5079
sacados/5079
Polynesie 2012 5 points
chapExoCorrec/5019
sacados/5019
Asie Juin 2012 5 points
-3
-2
-1
2
3
4
5
6
I
-1
2
3
4
5
6
7
J
O
C
f
E.3836
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
an
y
natural
n
um
b
er
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,
for
an
y
natural
n
um
b
er
n
:
v
n
=
u
n
+1
−
1
2
·
u
n
a
Calculate
v
0
.
b
Express
v
n
+1
as
a
function
of
v
n
.
c
Deduce
that
v
n
is
geometric
of
reason
1
2
.
d
Express
v
n
in
terms
of
n
.
3
W
e
define
the
sequence
w
n
b
y
p
osing,
for
an
y
natural
n
um
b
er
n
:
w
n
=
u
n
v
n
a
Calculate
w
0
.
b
Using
the
equalit
y
u
n
+1
=
v
n
+
1
2
·
u
n
,
express
w
n
+1
as
a
function
of
u
n
and
v
n
.
c
Deduce
that
for
an
y
n
of
N
:
w
n
+1
=
w
n
+2
.
d
Express
w
n
as
a
function
of
n
.
4
Sho
w
that
for
an
y
natural
n
um
b
er
n
:
u
n
=
2
·
n
−
1
2
n
5
F
or
an
y
natural
n
um
b
er
n
,
w
e
p
ose
:
S
n
=
k
=
n
k
=0
u
k
=
u
0
+
u
1
+
·
·
·
+
u
n
Demonstrate
b
y
recurrence
that
for
an
y
n
∈
N
:
S
n
=
2
−
2
n
+
3
2
n
E.3210
The
sequence
u
n
is
defined
b
y
:
u
0
=
1
;
u
n
+1
=
1
2
u
n
+
n
−
1
for
all
n
∈
N
1
a
Sho
w
that
for
an
y
n
3
:
u
n
0
.
b
Deduce
that
for
an
y
n
4
:
u
n
n
−
2
.
c
Deduce
the
limit
of
the
sequence
(
u
n
)
.
2
W
e
define
the
sequence
(
v
n
)
b
y
:
v
n
=
4
u
n
−
8
n
+
24
a
Sho
w
that
(
v
n
)
is
a
decreasing
geometric
sequence
whose
reason
and
first
term
will
b
e
giv
en.
b
Sho
w
that
for
an
y
natural
n
um
b
er
n
:
u
n
=
7
·
1
2
n
+
2
n
−
6
.
c
V
erify
that
for
an
y
natural
n
um
b
er
n
,
u
n
=
x
n
+
y
n
où
(
x
n
)
is
a
geometric
sequence
and
(
y
n
)
an
arithmetic
se-
quence
whose
first
term
and
reason
are
to
b
e
sp
ecified
for
eac
h.
d
Deduce
the
expression
of
S
n
=
n
k
=0
u
k
as
a
function
of
n
.
E.3472
Consider
the
sequence
u
n
n
∈
N
defined
b
y
:
u
0
=
1
u
n
+1
=
1
3
·
u
n
+
n
−
2
for
an
y
natural
n
um
b
er
n
.
1
Calculate
u
1
,
u
2
and
u
3
.
2
a
Demonstrate
that
for
an
y
natural
in
teger
n
4
:
u
n
0
.
b
Deduce
that
for
an
y
natural
n
um
b
er
n
5
:
u
n
n
−
3
c
Deduce
the
limit
of
the
sequence
u
n
n
∈
N
3
W
e
define
the
sequence
v
n
n
∈
N
b
y
:
F
or
an
y
n
∈
N
:
v
n
=
−
2
·
u
n
+
3
·
n
−
21
2
.
a
Sho
w
that
the
sequence
v
n
n
∈
N
is
a
geometric
se-
quence
whose
reason
and
first
term
will
b
e
giv
en.
b
Deduce
that
:
for
an
y
n
∈
N
:
u
n
=
25
4
·
1
3
n
+
3
2
·
n
−
21
4
.
c
Let
the
sum
S
n
b
e
defined
for
an
y
natural
in
teger
n
b
y
:
S
n
=
n
k
=0
u
k
Determine
the
expression
of
S
n
as
a
function
of
n
.
2.
Suites,
variations
and
limits
E.3415
Consider
the
function
f
defined
on
−∞
;
6
b
y
:
f
(
x
)
=
9
6
−
x
W
e
define
the
sequence
U
n
b
y
:
U
0
=
−
3
;
U
n
+1
=
f
U
n
for
an
y
n
∈
N
1
The
represen
tativ
e
curv
e
of
the
function
f
is
giv
en
b
e-
lo
w
accompanied
b
y
that
of
the
straigh
t
line
of
equation
y
=
x
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3836
sacados/3836
Antilles-Guyanne Septembre 2010 5 points
chapExoCorrec/3210
sacados/3210
Antilles-Guyane Septembre 2005 5 points
chapExoCorrec/3472
sacados/3472
chapExoCorrec/3415
sacados/3415
-3
-2
-1
2
3
4
5
6
I
-1
2
3
4
5
6
7
J
O
C
f
Construct,
in
the
ab
o
v
e
reference
frame,
the
p
oin
ts
:
M
0
(
U
0
;
0
)
;
M
1
(
U
1
;
0
)
;
M
2
(
U
2
;
0
)
;
M
3
(
U
3
;
0
)
;
M
4
(
U
4
;
0
)
What
conjectures
can
b
e
made
regarding
the
direction
of
v
ariation
and
p
ossible
con
v
ergence
of
the
sequence
U
n
?
2
a
Demonstrate
that
si
x
<
3
then
9
6
−
x
<
3
.
Deduce
that
U
n
<
3
for
an
y
natural
n
um
b
er
n
.
b
Study
the
direction
of
v
ariation
of
the
sequence
U
n
.
c
What
can
b
e
deduced
from
questions
2
a
and
2
b
?
3
Consider
the
sequence
V
n
defined
b
y
V
n
=
1
U
n
−
3
for
an
y
natural
n
um
b
er
n
.
a
Demonstrate
that
the
sequence
V
n
is
an
arithmetic
sequence
of
reason
−
1
3
.
b
Determine
V
n
then
U
n
as
a
function
of
n
.
c
Calculate
the
limit
of
the
sequence
U
n
.
E.5864
Consider
the
sequence
u
n
de-
fined
b
y
u
0
=
1
2
and
suc
h
that
for
an
y
natural
n
um
b
er
n
:
u
n
+1
=
3
·
u
n
1
+
2
·
u
n
1
a
Calculate
u
1
and
u
2
.
b
Demonstrate,
b
y
recurrence,
that
for
an
y
natural
n
um-
b
er
n
:
0
<
u
n
.
2
W
e
admit
that
for
an
y
natural
n
um
b
er
n
:
u
n
<
1
.
a
Demonstrate
that
the
sequence
u
n
is
increasing.
b
Demonstrate
that
the
sequence
u
n
con
v
erges.
3
Let
v
n
b
e
the
sequence
defined,
for
an
y
natural
n
um
b
er
n
b
y
:
v
n
=
u
n
1
−
u
n
a
Sho
w
that
the
sequence
v
n
is
a
geometric
sequence
of
reason
3
.
b
Express
for
an
y
natural
n
um
b
er
n
,
v
n
as
a
function
of
n
.
c
Deduce
that,
for
an
y
natural
n
um
b
er
n
:
u
n
=
3
n
3
n
+
1
d
Determine
the
limit
of
the
sequence
u
n
.
E.3474
Let
f
b
e
the
function
defined
on
the
in
terv
al
0
;
+
∞
b
y
:
f
(
x
)
=
6
−
5
x
+
1
The
aim
of
this
exercise
is
to
study
sequences
u
n
defined
b
y
a
first
p
ositiv
e
or
zero
term
u
0
and
v
erifying
for
an
y
natural
n
um
b
er
n
:
u
n
+1
=
f
(
u
n
)
1
Study
of
function
prop
erties
f
:
a
Study
the
direction
of
v
ariation
of
the
function
f
on
the
in
terv
al
0
;
+
∞
.
b
Solv
e
in
the
in
terv
al
0
;
+
∞
the
equation
:
f
(
x
)
=
x
.
Note
¸
the
solution.
c
Sho
w
that
if
x
b
elongs
to
the
in
terv
al
0
;
¸
,
then
f
(
x
)
b
elongs
to
the
in
terv
al
0
;
¸
.
Similarly
,
sho
w
that
if
x
b
elongs
to
the
in
terv
al
¸
;
+
∞
,
then
f
(
x
)
b
elongs
to
the
in
terv
al
¸
;
+
∞
.
2
Study
the
sequence
u
n
for
u
0
=
0
:
In
this
question,
consider
the
sequence
u
n
defined
b
y
u
0
=
0
and
for
an
y
natural
n
um
b
er
n
:
u
n
+1
=
f
u
n
=
6
−
5
u
n
+
1
a
On
the
graph
sho
wn
in
App
endix
2,
are
represen
ted
the
curv
es
with
equations
y
=
x
and
y
=
f
(
x
)
.
Place
the
p
oin
t
A
0
of
co
ordinates
(
u
0
;
0
)
,
and,
using
these
curv
es,
construct
from
A
0
the
p
oin
ts
A
1
,
A
2
,
A
3
and
A
4
with
ordinate
zero
and
abscissa
u
1
,
u
2
,
u
3
and
u
4
resp
ectiv
ely
.
What
conjectures
can
b
e
made
ab
out
the
direction
of
v
ariation
and
con
v
ergence
of
the
sequence
u
n
?
b
Demonstrate,
b
y
recurrence,
that
for
an
y
natural
n
um-
b
er
:
0
u
n
u
n
+1
¸
.
c
Deduce
that
the
sequence
u
n
is
con
v
ergen
t
and
de-
termine
its
limit.
3
Study
the
sequences
u
n
according
to
the
v
alues
of
the
p
ositiv
e
or
zero
real
u
0
.
In
this
question,
an
y
trace
of
argumen
tation,
ho
w
ev
er
incomplete,
or
initiativ
e,
ho
w
ev
er
unsuccessful,
will
b
e
tak
en
in
to
accoun
t
in
the
assessmen
t.
What
can
b
e
said
ab
out
the
direction
of
v
ariation
and
con
v
ergence
of
the
sequence
u
n
dep
ending
on
the
v
al-
ues
of
the
p
ositiv
e
or
zero
real
u
0
?
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5864
sacados/5864
chapExoCorrec/3474
sacados/3474
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4
5
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2
3
4
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E.6269
Consider
the
n
umerical
sequence
u
n
defined
on
N
b
y
:
u
0
=
2
;
u
n
+1
=
−
1
2
·
u
2
n
+
3
·
u
n
−
3
2
p
our
tout
n
∈
N
P
art
A
:
Conjecture
1
Calculate
the
exact
v
alues,
giv
en
in
irreducible
fractions,
of
u
1
and
u
2
.
2
Giv
e
an
appro
ximate
v
alue
to
the
nearest
10
−
5
of
the
terms
u
3
and
u
4
.
3
Conjecture
the
direction
of
v
ariation
and
con
v
ergence
of
the
sequence
u
n
.
P
art
B
:
V
alidation
of
conjectures
Consider
the
n
umerical
sequence
v
n
defined
for
an
y
natural
n
um
b
er
n
,
b
y
:
v
n
=
u
n
−
3
1
Sho
w
that,
for
an
y
natural
n
um
b
er
n
:
v
n
+1
=
−
1
2
·
v
2
n
.
2
Demonstrate
b
y
recurrence
that,
for
an
y
natural
n
um
b
er
n
:
−
1
v
n
0
.
3
a
Demonstrate
that,
for
an
y
natural
n
um
b
er
n
:
v
n
+1
−
v
n
=
−
v
n
·
1
2
·
v
n
+
1
.
b
Deduce
the
direction
of
v
ariation
of
the
sequence
v
n
.
4
Wh
y
can
w
e
then
sa
y
that
the
sequence
v
n
con
v
erges?
5
W
e
note
‘
the
limit
of
the
sequence
v
n
.
W
e
admit
that
‘
b
elongs
to
the
in
terv
al
−
1
;
0
and
v
er-
ifies
the
equalit
y
:
‘
=
−
1
2
·
‘
2
.
Determine
the
v
alue
of
‘
.
6
Are
the
conjectures
made
in
part
A
v
alidated?
E.6763
P
art
A
Let
u
n
b
e
the
sequence
defined
b
y
its
first
term
u
0
and,
for
an
y
natural
n
um
b
er
n
,
b
y
the
relation
:
u
n
+1
=
a
·
u
n
+
b
(
a
and
b
r
e
al
non-zer
o
such
that
a
=
1
)
W
e
p
ose,
for
an
y
natural
n
um
b
er
n
:
v
n
=
u
n
−
b
1
−
a
1
Demonstrate
that,
the
sequence
v
n
is
geometric
of
rea-
son
a
.
2
Deduce
that
if
a
b
elongs
to
the
in
terv
al
−
1
;
1
,
then
the
sequence
u
n
has
limit
b
1
−
a
.
P
art
B
In
Marc
h
2015
,
Max
buys
a
green
plan
t
measuring
80
cm
.
He
is
advised
to
prune
it
ev
ery
y
ear,
in
Marc
h,
b
y
cutting
off
a
quarter
of
its
heigh
t.
The
plan
t
will
then
gro
w
b
y
30
cm
o
v
er
the
follo
wing
t
w
elv
e
mon
ths.
As
so
on
as
he
gets
home,
Max
prunes
his
plan
t.
1
Ho
w
tall
will
the
plan
t
b
e
in
Marc
h
2016
b
efore
Max
prunes
it?
2
F
or
an
y
natural
n
um
b
er
n
,
let
h
n
b
e
the
heigh
t
of
the
plan
t,
b
efore
it
is
pruned,
in
Marc
h
of
the
y
ear
(2015+
n
)
.
a
Justify
that,
for
an
y
natural
n
um
b
er
n
:
h
n
+1
=
0.75
·
h
n
+
30
b
Use
the
calculator
to
conjecture
the
direction
of
v
aria-
tion
of
the
sequence
h
n
.
Demonstrate
this
conjecture
(r
e
asoning
by
r
e
curr
enc
e
may
b
e
use
d)
.
c
Is
the
sequence
h
n
con
v
ergen
t?
Justify
the
answ
er.
h
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hingmath.fr
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3
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7
8
9
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chapExoCorrec/6269
sacados/6269
chapExoCorrec/6763
sacados/6763
K
W
U
V
0
1
2
E.6962
The
aim
of
this
exercise
is
to
study
sequences
of
p
ositiv
e
terms
whose
first
term
u
0
is
strictly
greater
than
1
and
p
ossessing
the
follo
wing
prop
ert
y
:
for
an
y
natural
n
um
b
er
n
>
0
,
the
sum
of
the
first
n
consecutiv
e
terms
is
equal
to
the
pro
duct
of
the
first
n
consecutiv
e
terms.
W
e
admit
that
suc
h
a
sequence
exists
and
note
it
u
n
.
It
therefore
v
erifies
three
prop
erties
:
u
0
>
1
for
an
y
n
0
,
u
n
0
for
all
n
>
0
:
u
0
+
u
1
+
···
+
u
n
−
1
=
u
0
×
u
1
×···
×
u
n
−
1
1
W
e
c
ho
ose
u
0
=
3
.
Determine
u
1
and
u
2
.
2
F
or
an
y
in
teger
n
>
0
,
w
e
note
:
s
n
=
u
0
+
u
1
+
···
+
u
n
−
1
=
u
0
×
u
1
×···
×
u
n
−
1
.
In
particular,
w
e
ha
v
e
s
1
=
u
0
.
a
V
erify
that
for
an
y
in
teger
n
>
0
:
s
n
+1
=
s
n
+
u
n
and
s
n
>
1
b
Deduce
that
for
an
y
in
teger
n
>
0
:
u
n
=
s
n
s
n
−
1
c
Sho
w
that
for
an
y
n
0
:
u
n
>
1
.
3
The
function
f
b
elo
w
is
desiredb
elo
w
returns
the
v
alue
of
one
of
the
terms
in
the
sequence
u
n
où
the
argumen
ts
u
and
n
of
the
function
are
resp
ectiv
ely
the
first
term
of
the
sequence
u
n
and
the
rank
of
the
desired
term.
Function
f(u,n)
s
←
u
For
i
from
1
to
n
u
←
...
s
←
...
End
for
Return
u
a
Cop
y
and
complete
the
pro
cessing
part
of
the
ab
o
v
e
algorithm.
b
The
table
b
elo
w
giv
es
v
alues
rounded
to
the
thou-
sandth
of
u
n
for
differen
t
v
alues
of
the
in
teger
n
:
n
0
5
10
20
30
40
u
n
3
1.140
1.079
1.043
1.030
1.023
What
conjecture
can
b
e
made
ab
out
the
con
v
ergence
of
the
sequence
u
n
?
4
a
Justify
that
for
an
y
natural
n
um
b
er
n
>
0
:
s
n
>
n
.
b
Deduce
the
limit
of
the
sequence
s
n
and
then
that
of
the
sequence
u
n
.
E.5854
Let
t
w
o
sequences
u
n
and
v
n
b
e
defined
b
y
u
0
=
2
and
v
0
=
10
and
for
an
y
natural
n
um
b
er
n
:
u
n
+1
=
2
·
u
n
+
v
n
3
;
v
n
+1
=
u
n
+
3
·
v
n
4
P
art
A
Consider
the
function
f
ex-
tracted
from
an
algorithm
où
its
argumen
t
n
is
an
in
teger
greater
than
or
equal
to
1
:
W
e
call
the
function
f
with
the
v
alue
n
=
2
as
its
argumen
t.
Cop
y
and
complete
the
table
giv
en
b
e-
lo
w
giving
the
state
of
the
v
ari-
ables
during
this
call
:
Function
f(n)
K
←
0
U
←
2
V
←
10
As
long
as
K
<
n
K
←
K+1
W
←
U
U
←
2
·
U+V
3
V
←
W
+
3
·
V
4
End
as
long
as
Renvoyer
(
U
;
V
)
P
art
B
1
a
Sho
w
that
for
an
y
natural
n
um
b
er
n
:
v
n
+1
−
u
n
+1
=
5
12
·
v
n
−
u
n
b
F
or
an
y
natural
n
um
b
er
n
,
w
e
p
ose
:
w
n
=
v
n
−
u
n
.
Sho
w
that
for
an
y
natural
n
um
b
er
n
:
w
n
=
8
·
5
12
n
.
2
a
Sho
w
that
the
sequence
u
n
is
increasing
and
that
the
sequence
v
n
is
decreasing.
b
Deduce
from
the
results
of
questions
1
b
and
2
a
that
for
an
y
natural
n
um
b
er
n
,
w
e
ha
v
e
:
u
n
10
;
v
n
2
c
Deduce
that
the
sequences
u
n
and
v
n
are
con
v
er-
gen
t.
3
Sho
w
that
the
suites
u
n
and
v
n
ha
v
e
the
same
limit.
4
Sho
w
that
the
sequence
t
n
defined
b
y
t
n
=
3
·
u
n
+4
·
v
n
is
constan
t.
Deduce
that
the
common
limit
of
the
sequences
u
n
and
v
n
is
46
7
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6962
sacados/6962
chapExoCorrec/5854
sacados/5854
K
W
U
V
0
1
2
E.5077
The
purp
ose
of
this
exercise
is
to
study
the
sequence
u
n
defined
b
y
:
u
0
=
3
;
u
n
+1
=
1
2
·
u
n
+
7
u
n
p
our
tout
n
∈
N
W
e
ma
y
use
without
demonstration
the
fact
that
for
an
y
nat-
ural
n
um
b
er
n
,
u
n
>
0
.
1
W
e
denote
b
y
f
the
fo
cn
tion
defined
on
the
in
terv
al
0
;
+
∞
b
y
:
f
(
x
)
=
1
2
·
x
+
7
x
(
∗
)
Sho
w
that
the
function
f
admits
a
minim
um.
Deduce
that
for
an
y
natural
n
um
b
er
n
:
u
n
√
7
2
a
Let
n
b
e
an
y
natural
n
um
b
er.
Study
the
sign
of
u
n
+1
−
u
n
.
b
Wh
y
can
w
e
deduce
that
the
sequence
u
n
is
con
v
er-
gen
t?
c
F
rom
the
relation
(
∗
)
w
e
deduce
that
the
limit
‘
of
this
sequence
is
suc
h
that
:
‘
=
1
2
·
‘
+
7
‘
.
Determine
‘
.
3
Demonstrate
that
for
an
y
natural
n
um
b
er
n
:
u
n
+1
−
7
=
1
2
·
u
n
−
7
2
u
n
4
W
e
define
the
sequence
d
n
b
y
:
d
0
=
1
;
d
n
+1
=
1
2
·
d
n
2
p
our
tout
n
∈
N
a
Demonstrate
b
y
recurrence
that
for
an
y
natural
n
um-
b
er
n
:
u
n
−
7
d
n
b
Here’s
a
function
from
an
algorithm
taking
as
argu-
men
t
p
whic
h
is
a
natural
in
teger.
Function
f(p)
d
←
1
n
←
0
As
long
as
d
>
10
−
p
d
←
0.5
·
d
2
n
←
n+1
End
As
long
as
Return
n
Calling
the
function
with
the
v
alue
9
,
the
v
alue
re-
turned
is
the
n
um
b
er
5
.
What
inequalit
y
can
b
e
deduced
for
d
5
?
?
Justify
that
u
5
is
an
appro
ximate
v
alue
of
7
to
the
nearest
10
−
9
.
3.
Suites
and
lo
garithms
E.5966
P
art
A
Consider
the
sequence
u
n
defined
b
y
:
u
0
=
2
;
u
n
+1
=
1
+
3
u
n
3
+
u
n
for
an
y
natural
n
um
b
er
n
All
terms
in
this
sequence
are
assumed
to
b
e
definite
and
strictly
p
ositiv
e.
1
Demonstrate
b
y
recurrence,
that
for
an
y
natural
n
um
b
er
n
,
w
e
ha
v
e
:
u
n
>
1
2
a
Establish
that,
for
an
y
natural
n
um
b
er
n
,
w
e
ha
v
e
:
u
n
+1
−
u
n
=
(1
−
u
n
)(1
+
u
n
)
3
+
u
n
b
Determine
the
direction
of
v
ariation
of
the
sequence
u
n
.
Deduce
that
the
sequence
u
n
con
v
erges.
P
art
B
Consider
the
sequence
u
n
defined
b
y
:
u
0
=
2
;
u
n
+1
=
1
+
0.5
u
n
0.5
+
u
n
for
an
y
natural
n
um
b
er
n
.
All
terms
in
this
sequence
are
assumed
to
b
e
definite
and
strictly
p
ositiv
e.
1
Consider
the
follo
wing
algorithm
:
u
←
2
For
i
ranging
from
1
to
n
u
←
1+0.5u
0.5+u
End
For
Repro
duce
and
complete
the
table
b
elo
w
sho
wing
the
v
al-
ues
tak
en
b
y
the
v
ariables
i
and
u
during
the
step-b
y-step
execution
of
this
algorithm.
V
alues
of
u
will
b
e
rounded
to
the
thousandth.
i
1
2
3
u
2
F
or
n
=
12
,
w
e
extended
the
previous
table
and
obtained
:
i
4
5
6
7
8
9
10
11
12
u
1.0083
0.9973
1.0009
0.9997
1.0001
0.99997
1.00001
0.999996
1.000001
Conjecture
the
b
eha
vior
of
the
sequence
u
n
at
infinit
y
.
3
Consider
the
sequence
v
n
defined,
for
an
y
natural
n
um-
b
er
n
b
y
:
v
n
=
u
n
−
1
u
n
+
1
a
Demonstrate
that
the
sequence
v
n
is
geometric
of
reason
−
1
3
.
b
Calculate
v
0
then
write
v
n
as
a
function
of
n
.
4
a
Sho
w
that,
for
an
y
natural
n
um
b
er
n
,
w
e
ha
v
e
:
v
n
=
1
.
b
Sho
w
that,
for
an
y
natural
n
um
b
er
n
,
w
e
ha
v
e
:
u
n
=
1
+
v
n
1
−
v
n
c
Determine
the
limit
of
the
sequence
u
n
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5077
sacados/5077
Metropole Septembre 2012 5 points
chapExoCorrec/5966
sacados/5966
E.6264
A
patien
t
is
giv
en
a
drug
b
y
in
tra-
v
enous
injection.
The
quan
tit
y
of
drug
in
the
blo
o
d
decreases
as
a
function
of
time.
The
aim
of
the
exercise
is
to
study
,
for
differen
t
h
yp
otheses,
the
ev
olution
of
this
quan
tit
y
min
ute
b
y
min
ute.
1
A
t
time
0
an
injection
of
10
m‘
of
drug
is
made.
It
is
estimated
that
20
%
of
the
drug
is
eliminated
p
er
min
ute.
F
or
an
y
natural
n
um
b
er
n
,
note
u
n
the
amoun
t
of
drug,
in
m‘
,
remaining
in
the
blo
o
d
after
n
min
utes.
Th
us,
u
0
=
10
.
a
What
is
the
nature
of
the
sequence
u
n
?
b
F
or
an
y
natural
n
um
b
er
n
,
giv
e
the
expression
of
u
n
as
a
function
of
n
.
c
After
ho
w
long
do
es
the
amoun
t
of
drug
remaining
in
the
blo
o
d
b
ecome
less
than
1
%
of
the
initial
amoun
t?
Justify
the
answ
er.
2
A
t
time
0
,
a
mac
hine
injects
10
m‘
of
drug.
It
is
esti-
mated
that
20
%
of
the
drug
is
eliminated
p
er
min
ute.
When
the
amoun
t
of
drug
falls
b
elo
w
5
m‘
,
the
mac
hine
re-injects
4
m‘
of
pro
duct.
After
15
min
utes,
the
mac
hine
is
stopp
ed.
F
or
an
y
natural
n
um
b
er
n
,
note
v
n
the
amoun
t
of
drug,
in
m‘
,
remaining
in
the
blo
o
d
at
min
ute
n
.
The
follo
wing
algorithm
mak
es
it
p
ossible,
through
the
v
alues
tak
en
b
y
the
v
ariable
v
,
to
obtain
the
remaining
quan
tit
y
of
drug
min
ute
b
y
min
ute.
v
←
10
For
n
ranging
from
1
to
15
v
←
0.8
×
V
If
v<5
Then
v
←
v+4
End
If
End
For
a
Calculate
the
missing
elemen
ts
of
the
table
b
elo
w
giv-
ing,
rounded
to
10
−
2
and
for
n
greater
than
or
equal
to
1
,
the
remaining
min
ute-b
y-min
ute
quan
tit
y
of
drug
obtained
with
the
algorithm.
n
0
1
2
3
4
5
6
7
v
n
10
8
6.4
8.15
n
8
9
10
11
12
13
14
15
v
n
6.52
5.21
8.17
6.54
5.23
8.18
6.55
5.24
b
After
15
min
utes,
what
total
amoun
t
of
drug
has
b
een
injected
in
to
the
b
o
dy?
c
W
e
w
an
t
to
program
the
mac
hine
to
inject
2
m‘
pro
d-
uct
when
the
amoun
t
of
drug
in
the
blo
o
d
is
less
than
or
equal
to
6
m‘
and
that
it
stops
after
30
min
utes.
Recop
y
the
previous
algorithm,
mo
difying
it
so
that,
at
the
end
of
execution,
the
v
alue
of
v
ariable
b
is
the
quan
tit
y
of
drug,
in
m‘
,
remaining
in
the
blo
o
d
min
ute
b
y
min
ute
with
this
new
proto
col.
3
The
mac
hine
is
programmed
so
that
:
at
time
0
,
she
injects
10
m‘
of
medication,
ev
ery
min
ute,
she
injects
1
m‘
of
medication.
It
is
estimated
that
20
%
of
the
drug
in
the
blo
o
d
is
elim-
inated
p
er
min
ute.
F
or
an
y
natural
n
um
b
er
n
,
note
w
n
the
amoun
t
of
drug,
in
m‘
,
presen
t
in
the
patien
t’s
blo
o
d
after
n
min
utes.
a
Justify
that
for
an
y
natural
n
um
b
er
n
:
w
n
+1
=
0.8
·
w
n
+
1
.
b
F
or
an
y
natural
n
um
b
er
n
,
w
e
p
ose
:
z
n
=
w
n
−
5
.
Sho
w
that
z
n
is
a
geometric
sequence
whose
reason
and
first
term
are
to
b
e
sp
ecified.
c
Deduce
the
expression
of
w
n
as
a
function
of
n
.
d
What
is
the
limit
of
the
sequence
w
n
?
What
in
ter-
pretation
can
b
e
giv
en?
E.3234
F
or
an
y
natural
n
um
b
er
n
,
w
e
p
ose
u
n
=
n
10
2
n
.
This
defines
a
sequence
u
n
n
∈
N
1
Pro
v
e,
for
an
y
non-zero
natural
n
um
b
er
n
,
the
follo
wing
equiv
alence
:
u
n
+1
0.95
u
n
if,
and
only
if,
1+
1
n
10
1.9
2
Consider
the
function
f
defined
on
1
;
+
∞
b
y
:
f
(
x
)
=
1
+
1
x
10
a
Study
the
direction
of
v
ariation
and
limit
in
+
∞
of
the
function
f
.
b
Sho
w
that
there
exists
in
the
in
terv
al
1
;
+
∞
a
single
real
n
um
b
er
¸
suc
h
that
f
(
¸
)
=
1.9
.
c
Determine
the
natural
n
um
b
er
n
0
suc
h
that
n
0
−
1
¸
n
0
.
d
Sho
w
that,
for
an
y
natural
n
um
b
er
n
greater
than
or
equal
to
16
,
w
e
ha
v
e
:
1
+
1
n
10
1.9
.
3
a
Determine
the
direction
of
v
ariation
of
the
sequence
(
u
n
)
from
rank
16
.
b
What
can
w
e
deduce
for
the
sequence?
4
Using
reasoning
b
y
recurrence,
pro
v
e,
for
an
y
natural
n
um
b
er
n
greater
than
or
equal
to
16,
the
framew
ork
:
0
u
n
0.95
n
−
16
·
u
16
Deduce
the
limit
of
the
sequence
u
n
n
∈
N
E.3267
1
Let
u
b
e
the
sequence
defined
b
y
:
u
0
=
0
u
n
+1
=
1
2
−
u
n
for
an
y
natural
n
um
b
er
n
.
a
Calculate
u
1
,
u
2
and
u
3
.
Eac
h
of
these
terms
will
b
e
expressed
as
an
irreducible
fraction.
b
Compare
the
first
four
terms
of
the
sequence
u
with
the
first
four
terms
of
the
sequence
w
defined
on
N
b
y
:
w
n
=
n
n
+
1
.
c
Using
reasoning
b
y
recurrence,
sho
w
that,
for
an
y
nat-
ural
n
um
b
er
n
:
u
n
=
w
n
.
2
Let
v
b
e
the
sequence
with
general
term
v
n
defined
b
y
:
v
n
=
ln
n
n
+
1
où
ln
denotes
the
nep
erian
logarithm
function
:
a
Sho
w
that
:
v
1
+
v
2
+
v
3
=
−
ln
4
b
Let
S
n
b
e
the
sum
defined
for
an
y
non-zero
natural
n
um
b
er
n
b
y
:
S
n
=
v
1
+
v
2
+
·
·
·
+
v
n
Express
S
n
as
a
function
of
n
.
Determine
the
limit
of
S
n
when
n
tends
to
+
∞
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6264
sacados/6264
chapExoCorrec/3234
sacados/3234
chapExoCorrec/3267
sacados/3267
4.
Suites
and
inte
gr
ations
E.3161
Consider
the
sequences
u
n
and
v
n
defined,
for
an
y
non-zero
natural
n
um
b
er
n
,
b
y
:
u
1
=
1
u
n
=
u
n
−
1
+
1
n
p
our
n
2
;
v
n
=
u
n
−
ln
n
for
n
1
1
a
Calculate
u
2
,
u
3
and
u
4
.
b
Sho
w
that,
for
an
y
non-zero
natural
n
um
b
er
n
:
u
n
=
n
k
=1
1
k
2
a
Sho
w
that,
for
an
y
non-zero
natural
n
um
b
er
k
:
1
k
+
1
k
+1
k
1
x
d
x
1
k
b
Deduce
that,
for
an
y
in
teger
n
greater
than
or
equal
to
2
,
w
e
ha
v
e
the
follo
wing
inequalities
:
u
n
−
1
ln
n
u
n
−
1
n
;
0
v
n
1
3
a
Sho
w
that,
for
an
y
non-zero
natural
n
um
b
er
n
:
v
n
+1
−
v
n
=
1
n
+
1
−
n
+1
n
1
x
d
x
b
Deduce
the
direction
of
v
ariations
of
the
sequence
(
v
n
)
.
4
Sho
w
that
the
sequence
(
v
n
)
con
v
erges.
W
e
note
‚
the
limit
of
the
sequence
(
v
n
)
(we
wil
l
not
try
to
c
alculate
‚
)
.
What
is
the
limit
of
the
sequence
(
u
n
)
?
E.3144
1
F
or
an
y
non-zero
natural
n
um
b
er
n
,
consider
the
func-
tion
f
n
defined
on
0
;
+
∞
b
y
:
f
n
(
x
)
=
ln
x
+
x
n
−
1
a
Determine
the
limits
of
f
n
at
0
and
+
∞
then
study
the
direction
of
v
ariations
of
f
n
.
b
Sho
w
that
the
equation
f
n
(
x
)
=
0
admits
a
single
solu-
tion
in
0
;
+
∞
.
Note
¸
n
this
solution.
Sho
w
that
it
b
elongs
to
the
in
terv
al
1
;
e
.
2
The
plane
is
referred
to
an
orthonormal
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
.
W
e
denote
(Γ)
the
represen
tativ
e
curv
e
of
the
nep
erian
logarithm
function.
a
Let
n
b
e
a
non-zero
natural
n
um
b
er.
Determine
an
equation
of
the
line
Δ
n
passing
through
the
p
oin
t
A
of
co
ordinates
(
0
;
1
)
and
the
p
oin
t
B
n
of
co
ordinates
(
n
;
0
)
.
b
Mak
e
a
sk
etc
h
represen
ting
the
curv
e
(Γ)
and
the
straigh
t
lines
Δ
1
,
Δ
2
and
Δ
3
.
c
Sho
w
that
¸
n
is
the
abscissa
of
the
p
oin
t
of
in
tersection
of
(Γ)
with
Δ
n
.
d
Sp
ecify
the
v
alue
of
¸
1
then
mak
e
a
conjecture
ab
out
the
direction
of
v
ariation
of
the
sequence
¸
n
)
.
3
a
Express
ln
(
¸
n
)
as
a
function
of
n
and
¸
n
.
b
Express
f
n
+1
(
¸
n
)
as
a
function
of
n
and
¸
n
and
c
hec
k
that
:
f
n
+1
(
¸
n
)
<
0
c
Deduce
from
the
previous
question
the
direction
of
v
ariation
of
the
sequence
(
¸
n
)
.
d
Sho
w
that
the
sequence
(
¸
n
)
con
v
erges.
Let
‘
b
e
its
limit.
Establish
that
ln
‘
=
1
and
deduce
the
v
alue
of
‘
.
4
W
e
denote
b
y
D
n
the
domain
b
ounded
b
y
the
curv
e
(Γ)
,
the
x-axis
and
the
straigh
t
lines
of
equation
:
x
=
¸
n
and
x
=
e
.
a
Calculate
the
area
of
the
domain
D
n
as
a
function
of
¸
n
and
sho
w
that
this
area
is
equal
to
¸
2
n
n
.
b
Establish
that
:
(e
−
¸
n
)ln
¸
n
¸
2
n
n
(e
−
¸
n
)
c
Deduce
a
frame
for
n
(e
−
¸
n
)
.
d
is
the
general
term
sequence
n
(e
−
¸
n
)
con
v
ergen
t?
Do
es
this
result
allo
w
us
to
appreciate
the
sp
eed
of
con
v
ergence
of
the
sequence
(
¸
n
)
?
5.
Unclassifie
d
financial
ye
ars
E.8144
P
olynesia
June
2018
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3161
sacados/3161
chapExoCorrec/3144
sacados/3144
sacados/8144
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