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Grade 11
/ Study of sequences
66 exercises (including 58 corrected)
a
2
3
4
5
6
7
8
I
-1
2
3
J
O
C
f
ChingQuizz
:
7
exercises
a
v
ailable
for
Quizz
assessmen
t
:
1.
V
ariations
of
the
first
terms
of
arithmetic
and
ge
ometric
se
quenc
es
E.6015
Definition
:
Let
u
n
b
e
a
sequence
defined
for
all
n
∈
N
.
The
sequence
u
n
is
said
to
b
e
strictly
increasing
if
for
all
n
∈
N
,
w
e
ha
v
e
:
u
n
+1
>
u
n
the
sequence
u
n
is
said
to
b
e
strictly
decreasing
if
for
all
n
∈
N
,
w
e
ha
v
e
:
u
n
+1
<
u
n
the
sequence
u
n
is
said
to
b
e
constan
t
if
for
all
n
∈
N
,
w
e
ha
v
e
:
u
n
+1
=
u
n
Consider
the
arithmetic
sequence
u
n
n
∈
N
with
first
term
5
and
common
difference
2
.
1
Giv
e
the
first
four
terms
of
the
sequence
u
n
.
2
Express
the
v
alue
of
the
term
u
n
in
terms
of
its
rank
n
.
3
What
conjecture
can
b
e
made
ab
out
the
v
ariation
of
the
terms
of
the
sequence
u
n
?
E.6016
Consider
the
geometric
sequence
u
n
n
∈
N
with
first
term
24
and
reason
1
2
.
1
Giv
e
the
first
four
terms
of
the
sequence
u
n
.
2
Giv
e
the
expression
of
the
general
term
of
the
sequence
u
n
as
a
function
of
its
rank
n
.
3
What
conjecture
can
b
e
made
ab
out
the
v
ariation
of
the
terms
of
the
sequence
u
n
?
E.6653
In
eac
h
case,
sp
ecify
,
if
p
ossible,
the
direction
of
v
ariation
of
the
sequences
:
1
u
n
is
an
arithmetic
sequence
whose
first
term
is
p
osi-
tiv
e
and
whose
reason
is
negativ
e.
2
v
n
is
a
geometric
sequence
whose
first
term
is
negativ
e
and
whose
reason
is
strictly
greater
than
1.
3
w
n
is
a
geometric
sequence
whose
first
term
is
p
ositiv
e
and
reason
is
negativ
e.
2.
V
ariations
of
the
first
terms
of
an
explicitly
define
d
se
quenc
e
E.5089
Consider
the
function
f
defined
on
R
+
whose
represen
tativ
e
curv
e
C
f
is
giv
en
in
the
orthonormal
frame
O
;
I
;
J
b
elo
w
:
W
e
define
the
sequence
u
n
b
y
the
relation
:
u
n
=
f
(
n
)
for
an
y
in
teger
n
∈
N
.
1
Justify
that
the
term
u
4
has
the
v
alue
3
2
.
2
a
Determine
the
v
alue
of
terms
:
u
0
;
u
1
;
u
2
;
u
3
;
u
4
;
u
5
;
u
6
b
Sa
y
whether
the
statemen
ts
b
elo
w
are
true
or
not
:
ˇ
The
terms
of
the
sequence
u
n
for
i
∈
0
;
1
;
2
are
ordered
in
descending
order.
ı
ˇ
The
terms
of
the
sequence
u
n
for
i
∈
3
;
4
;
5
are
ordered
in
ascending
order.
ı
E.1585
Consider
the
t
w
o
sequences
u
n
and
v
n
defined
for
an
y
natural
n
um
b
er
n
(
n
∈
N
)
:
u
n
=
2
·
n
2
−
n
+
1
;
v
n
=
4
−
n
1
+
n
1
Determine
the
first
5
terms
of
these
t
w
o
sequences.
2
Conjecture
the
direction
of
v
ariations
of
the
sequence
u
n
and
v
n
.
E.8477
Consider
the
t
w
o
sequences
u
n
and
v
n
defined
for
an
y
natural
n
um
b
er
n
(
n
∈
N
)
b
y
:
u
n
=
n
3
−
3
·
n
2
+
2
·
n
+
2
;
v
n
=
3
·
1
+
(
−
1)
n
+
2
1
Determine
the
first
5
terms
of
the
sequences
u
n
and
v
n
2
F
or
eac
h
of
the
conjectures,
sa
y
whether
they
are
proba-
ble
or
not
:
ˇ
the
sequence
u
n
is
constan
t.
ı
ˇ
the
follo
wing
v
n
is
constan
t.
ı
3.
V
ariations
of
the
first
terms
of
a
se
quenc
e
define
d
by
r
e
curr
enc
e
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6015
sacados/6015
chapExoCorrec/6016
sacados/6016
chapExoCorrec/6653
sacados/6653
chapExoCorrec/5089
sacados/5089
2
3
4
5
6
7
8
I
-1
2
3
J
O
C
f
chapExoCorrec/1585
sacados/1585
chapExoCorrec/8477
sacados/8477
-2
-1
2
3
4
I
-1
2
J
O
C
f
-4
-3
-2
-1
2
3
4
I
-3
-2
-1
2
3
J
O
C
f
E.2378
Consider
the
function
f
defined
on
[
−
2
;
4]
whose
represen
tativ
e
curv
e
C
f
is
giv
en
b
elo
w
:
Consider
the
sequence
u
n
n
∈
N
defined
b
y
the
relation
u
0
=
3
;
u
n
+1
=
f
u
n
for
all
n
∈
N
1
Complete
the
follo
wing
table
with
the
first
terms
of
the
sequence
u
n
:
n
0
1
2
3
4
5
6
7
8
9
u
n
2
F
or
eac
h
of
the
statemen
ts
b
elo
w,
state
whether
they
are
true
or
false
:
ˇ
the
sequence
u
n
is
strictly
decreasing
on
N
.
ı
ˇ
the
sequence
u
n
is
constan
t
from
rank
3
.
ı
E.8478
1
Consider
the
sequence
u
n
defined
for
an
y
natural
n
um-
b
er
n
(
n
∈
N
)
b
y
:
u
0
=
2
;
u
n
+1
=
1
2
·
u
n
−
1
4
a
Determine
the
4
first
terms
of
the
suite
u
n
.
b
Conjecture
the
v
ariation
of
the
sequence
u
n
2
Consider
the
sequence
v
n
defined
for
an
y
natural
n
um-
b
er
n
(
n
∈
N
)
b
y
:
v
0
=
−
1
;
v
n
+1
=
1
2
·
v
n
−
1
4
a
Justify
comparisons
:
v
0
<
v
1
<
v
2
<
v
3
b
Conjecture
the
v
ariation
of
the
sequence
v
n
E.8479
Consider
the
sequence
u
n
defined
for
an
y
natural
n
um
b
er
n
(
n
∈
N
)
b
y
:
u
0
=
1
;
u
n
+1
=
2
1
+
2
u
n
1
Determine
the
first
5
terms
of
the
suite
u
n
.
2
Conjecture
the
direction
of
v
ariations
of
the
sequence
u
n
.
3
Conjecture
the
explicit
expression
of
the
general
term
of
the
sequence
u
n
as
a
function
of
its
rank
n
.
4.
V
ariations
:
c
onstant
or
p
erio
dic
se
quenc
es
E.2978
In
the
plane
pro
vided
with
an
or-
thonormal
reference
frame
O
;
I
;
J
,
consider
the
represen
ta-
tion
C
f
of
a
function
f
defined
on
the
in
terv
al
[
−
4
;
4]
:
1
Consider
the
sequence
(
u
n
)
defined
for
an
y
natural
n
um-
b
er
n
(
n
∈
N
)
defined
b
y
:
u
0
=
4
;
u
n
+1
=
f
u
n
a
Determine
the
v
alue
of
the
first
6
terms
of
the
suite
u
n
.
b
Determine
the
v
alue
of
term
u
100
.
2
Consider
the
sequence
(
v
n
)
defined
for
an
y
natural
n
um-
b
er
n
(
n
∈
N
)
defined
b
y
:
v
0
=
−
2
;
v
n
+1
=
f
v
n
Determine
the
v
alue
of
the
term
v
100
.
E.2953
Consider
the
sequence
u
n
n
∈
N
de-
fined
b
y
the
follo
wing
recurrence
relation
:
u
0
=
3
;
u
n
+1
=
1
−
u
n
1
+
u
n
for
all
n
∈
N
1
Determine
the
first
fiv
e
terms
of
the
sequence
u
n
.
2
Sho
w
that
w
e
ha
v
e
the
follo
wing
relationship
:
u
n
+2
=
u
n
for
all
n
∈
N
3
What
can
b
e
said
ab
out
the
terms
of
this
sequence?
4
W
e
admit
that
the
term
of
rank
n
of
the
sequence
u
n
admits
an
expression
of
the
form
:
u
n
=
a
·
1
−
(
−
1)
n
+
b
p
our
tout
n
∈
N
where
a
and
b
are
t
w
o
real
n
um
b
ers
(
a;b
∈
R
)
.
Determine
the
v
alues
of
a
and
b
.
5.
V
ariations
:
explicit
se
quenc
es
h
ttps:/
/c
hingmath.fr
chapExoCorrec/2378
sacados/2378
-2
-1
2
3
4
I
-1
2
J
O
C
f
chapExoCorrec/8478
sacados/8478
chapExoCorrec/8479
sacados/8479
chapExoCorrec/2978
sacados/2978
-4
-3
-2
-1
2
3
4
I
-3
-2
-1
2
3
J
O
C
f
chapExoCorrec/2953
sacados/2953
u
n
est croissante
u
n
est décroissante
u
0
>
0
et
r>
0
u
0
>
0
et
r<
0
u
0
<
0
et
r>
0
u
0
<
0
et
r<
0
E.8485
The
sequence
u
n
n
∈
N
is
de-
fined
b
y
:
u
n
=
−
2
·
n
2
−
3
·
n
+
2
for
all
n
∈
N
Study
the
monotonicit
y
of
eac
h
of
the
sequences
b
elo
w,
b
y
studying
the
function
f
v
erifying
the
relation
:
u
n
=
f
(
n
)
for
all
n
∈
N
E.2386
Consider
the
sequence
u
n
n
∈
N
whose
term
of
rank
n
is
giv
en
b
y
the
form
ula
:
u
n
=
n
2
−
7
·
n
+
1
1
Using
the
calculator,
complete
the
table
b
elo
w
:
n
0
1
2
3
4
5
6
7
8
9
10
u
n
2
After
giving
the
table
of
v
ariations
of
the
function
f
whose
image
of
x
is
defined
b
y
:
f
(
x
)
=
x
2
−
7
·
x
+
1
Establish
that
the
sequence
u
n
is
increasing
from
rank
4.
6.
V
ariations
:
explicit
se
quenc
es
with
derivatives
E.2382
The
sequence
u
n
is
defined
b
y
:
u
n
=
2
·
n
2
+1
2
·
n
+5
for
all
n
∈
N
Consider
the
function
f
defined
b
y
:
f
(
x
)
=
2
·
x
2
+
1
2
·
x
+
5
1
Giv
e
the
defining
set
D
f
of
the
function
f
.
2
Establish
that
the
function
f
deriv
ed
from
the
function
f
admits
as
expression
on
D
f
:
f
(
x
)
=
4
·
x
2
+
20
·
x
−
2
(2
·
x
+
5)
2
3
Dra
w
up
the
table
of
v
ariations
of
the
function
f
.
4
Justify
that
the
sequence
u
n
is
increasing
from
rank
1
.
5
Can
w
e
sa
y
that
the
sequence
u
n
is
increasing
on
N
?
E.2410
Establish
the
monotonicit
y
on
N
of
the
sequence
u
n
n
∈
N
defined
b
y
the
explicit
form
ula
:
u
n
=
n
2
−
1
√
n
for
all
n
∈
N
.
7.
V
ariations
of
arithmetic
se
quenc
es
E.10424
Consider
the
suite
u
n
arithmetic,
defined
for
an
y
n
∈
N
,
with
first
term
of
2
and
reason
3
:
1
Giv
e
the
expression
of
the
terms
of
the
sequence
u
n
as
a
function
of
their
rank
n
.
2
F
or
n
∈
N
,
simplify
the
expression
u
n
+1
−
u
n
3
Deduce
the
direction
of
v
ariation
of
the
sequence
u
n
.
Prop
osition
:
let
u
n
b
e
an
arithmetic
sequence
:
u
n
is
strictly
decreasing
if,
and
only
if,
its
reason
is
strictly
negativ
e
;
u
n
is
strictly
increasing
if,
and
only
if,
its
reason
is
strictly
p
ositiv
e
;
u
n
is
constan
t
if,
and
only
if,
its
reason
is
zero
;
E.10425
Let
u
n
b
e
an
arithmetic
sequence
with
first
term
u
0
and
reason
r
.
1
Connect
the
corresp
onding
assertions
:
2
A
quelle
(s)
condition
(s)
arithmetic
sequence
is
constan
t?
E.8495
Let
u
n
n
∈
N
b
e
the
arithmetic
se-
quence
of
first
term
1
and
reason
3
.
Justify
that
the
sequence
u
n
is
an
increasing
sequence
on
N
.
E.10423
Let
v
n
n
∈
N
b
e
the
sequence
defined
for
an
y
natural
n
um
b
er
n
(
n
∈
N
)
and
whose
term
of
rank
n
admits
the
expression
:
v
n
=
4
−
n
Justify
that
the
sequence
v
n
is
decreasing
on
N
.
E.2380
Let
u
n
n
∈
N
b
e
the
sequence
whose
rank
term
n
is
defined
b
y
:
u
n
=
−
32
n
+
102
for
all
n
∈
N
Sho
w
that
this
sequence
is
decreasing.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8485
sacados/8485
fichierPlus/8485/diapoCorrection.pdf
chapExoCorrec/2386
sacados/2386
chapExoCorrec/2382
sacados/2382
fichierPlus/2382/
chapExoCorrec/2410
sacados/2410
sacados/10424
sacados/10425
u
n
est croissante
u
n
est décroissante
u
0
>
0
et
r>
0
u
0
>
0
et
r<
0
u
0
<
0
et
r>
0
u
0
<
0
et
r<
0
chapExoCorrec/8495
sacados/8495
chapExoCorrec/10423
sacados/10423
chapExoCorrec/2380
sacados/2380
0
<q<
1
q
q>
1
u
0
<
0
u
0
u
0
>
0
u
n
est croissante
u
n
est décroissante
u
0
>
0
et
q>
1
u
0
>
0
et
0
<q<
1
u
0
>
0
et
−
1
<q<
0
u
0
>
0
et
q<
−
1
u
0
<
0
et
q>
1
u
0
<
0
et
0
<q<
1
u
0
<
0
et
−
1
<q<
0
u
0
<
0
et
q<
−
1
8.
V
ariations
of
ge
ometric
se
quenc
es
E.10427
Consider
the
sequence
u
n
defined
for
all
n
∈
N
geometric
terms
with
first
term
3
and
common
ratio
0.1
.
1
Express
the
term
of
the
sequence
u
n
of
rank
n
in
terms
of
n
.
2
Simplify
and
factor
the
expression
u
n
+1
−
u
n
.
3
Deduce
the
direction
of
v
ariation
of
the
sequence
u
n
.
Prop
osition
:
Let
u
n
b
e
a
geometric
sequence
with
com-
mon
ratio
q
.
If
u
0
>
0
and
q
>
1
,
then
the
sequence
u
n
is
strictly
increasing.
If
q
=
1
,
then
the
sequence
u
n
is
constan
t.
If
u
0
>
0
and
0
<
q
<
1
,
then
the
sequence
u
n
is
strictly
decreasing.
E.9846
Consider
the
suite
u
n
n
∈
N
geomet-
ric
with
first
term
u
0
and
reason
q
.
In
the
table
b
elo
w,
indicate
the
direction
of
v
ariation
of
the
sequence
u
n
as
a
function
of
the
v
alue
of
its
c
haracteristic
elemen
ts
:
E.10426
Let
u
n
b
e
a
geometric
sequence
with
first
term
u
0
and
reason
r
.
1
Connect
the
corresp
onding
assertions
:
2
A
quelle
(s)
condition
(s)
geometric
sequence
is
constan
t?
9.
V
ariations
:
differ
enc
e
b
etwe
en
c
onse
cutive
terms
E.8480
Consider
the
sequence
u
n
defined
b
y
:
u
n
=
1
−
n
1
+
n
for
all
n
∈
N
1
Determine
a
simplified
expression
for
u
n
+1
−
u
n
.
2
Deduce
the
v
ariations
of
the
sequence
u
n
on
N
.
E.8482
Consider
the
sequence
u
n
defined
b
y
the
relation
:
u
n
=
n
3
−
4
·
n
2
+
n
−
3
for
all
n
∈
N
1
Establish
the
iden
tit
y
b
elo
w
for
an
y
natural
n
um
b
er
n
:
u
n
+1
−
u
n
=
3
·
n
2
−
5
·
n
−
2
.
2
Deduce
that
the
sequence
u
n
is
increasing
from
rank
2
.
E.8481
Let
u
n
n
∈
N
b
e
defined
b
y
the
ex-
plicit
relation
:
u
n
=
n
3
−
2
n
2
−
3
n
1
Giv
e
the
simplified
expression
of
:
u
n
+1
−
u
n
.
2
Deduce
that
the
sequence
(
u
n
)
is
increasing
for
n
greater
than
2.
E.3401
Consider
the
sequence
u
n
defined
b
y
:
u
n
=
n
2
+
10
2
·
n
for
an
y
n
∈
N
∗
Justify
that
u
n
is
increasing
from
rank
3
.
E.6040
Consider
the
sequence
u
n
defined
on
N
defined
b
y
:
u
n
=
5
n
−
1
(
n
+
1)
2
1
Giv
e
the
simplified
and
factorized
form
of
the
difference
:
u
n
+1
−
u
n
2
Justify
that
the
sequence
u
n
is
decreasing
from
rank
1
.
h
ttps:/
/c
hingmath.fr
sacados/10427
chapExoCorrec/9846
sacados/9846
0
<q<
1
q
q>
1
u
0
<
0
u
0
u
0
>
0
sacados/10426
u
n
est croissante
u
n
est décroissante
u
0
>
0
et
q>
1
u
0
>
0
et
0
<q<
1
u
0
>
0
et
−
1
<q<
0
u
0
>
0
et
q<
−
1
u
0
<
0
et
q>
1
u
0
<
0
et
0
<q<
1
u
0
<
0
et
−
1
<q<
0
u
0
<
0
et
q<
−
1
chapExoCorrec/8480
sacados/8480
chapExoCorrec/8482
sacados/8482
chapExoCorrec/8481
sacados/8481
chapExoCorrec/3401
sacados/3401
chapExoCorrec/6040
sacados/6040
E.8496
Let
u
n
b
e
the
sequence
whose
rank
term
n
is
defined
b
y
:
u
n
=
2
n
−
1
for
all
n
∈
N
∗
1
Establish
the
iden
tit
y
b
elo
w
for
an
y
strictly
p
ositiv
e
nat-
ural
n
um
b
er
n
(
n
∈
N
∗
)
:
u
n
+1
−
u
n
=
2
2
n
+
1
+
2
n
−
1
2
deduce
that
the
suite
u
n
is
strictly
increasing
on
N
∗
.
E.8497
Let
w
n
b
e
the
sequence
whose
rank
term
n
is
defined
b
y
:
w
n
=
2
n
−
25
n
for
all
n
∈
N
∗
Sho
w
that
the
sequence
w
n
is
increasing.
E.8483
The
sequence
u
n
is
defined
b
y
the
explicit
form
ula
:
u
n
=
5
+
n
n
for
all
n
∈
N
∗
Determine
the
direction
of
v
ariation
of
the
sequence
u
n
.
10.
V
ariations
:
differ
enc
e
of
c
onse
cutive
terms
(se
quenc
es
define
d
by
r
e
curr
enc
e)
E.5367
Let
u
n
n
∈
N
b
e
the
sequence
defined
b
y
:
u
0
=
1
;
u
n
+1
=
u
n
−
u
n
2
−
1
for
all
n
∈
N
1
Complete
the
table
b
elo
w
of
the
first
terms
of
the
se-
quence
u
n
:
n
0
1
2
3
4
u
n
2
By
studying
the
difference
of
t
w
o
consecutiv
e
terms,
sho
w
that
the
sequence
u
n
is
decreasing.
E.10639
Consider
the
sequence
u
n
defined
on
N
b
y
:
u
0
=
−
3
;
u
n
+1
=
u
n
−
3
n
+
7
1
Determine
the
first
four
terms
of
the
sequence
u
n
.
2
Determine
from
whic
h
rank
the
sequence
u
n
is
decreas-
ing.
11.
V
ariations
:
quotient
of
c
onse
cutive
terms
E.8494
1
Let
u
n
b
e
the
geometric
sequence
defined
at
N
with
first
term
2
and
reason
4
.
Justify
that
the
sequence
u
n
is
increasing.
2
Let
v
n
b
e
the
sequence
defined
for
an
y
natural
n
um
b
er
n
(
n
∈
N
)
and
whose
term
of
rank
n
admits
the
expres-
sion
:
v
n
=
3
×
0
;
2
n
Justify
that
the
sequence
v
n
is
decreasing.
E.2381
Consider
the
sequence
u
n
defined
b
y
:
u
n
=
3
n
4
for
an
y
n
∈
N
.
Sho
w
that
u
n
is
strictly
increasing.
E.2522
Consider
the
sequence
u
n
defined
for
an
y
natural
n
um
b
er
n
(
n
∈
N
)
b
y
:
u
n
=
5
n
n
+
2
Sho
w
that
the
sequence
u
n
n
∈
N
is
an
increasing
sequence
on
N
.
E.5491
Consider
the
sequence
u
n
defined
b
y
:
u
n
=
3
n
2
n
+
1
for
all
n
∈
N
Determine
the
v
ariations
of
the
sequence
u
n
on
N
.
E.10471
Consider
the
sequence
u
n
defined
on
N
b
y
:
u
n
=
2
n
n
2
+
1
Determine
from
whic
h
rank
the
sequence
u
n
is
increasing.
E.2451
The
sequence
u
n
is
defined
b
y
the
explicit
form
ula
:
u
n
=
2
n
3
·
n
for
all
n
∈
N
∗
Determine
the
direction
of
v
ariation
of
the
sequence
u
n
.
12.
V
ariations
fr
om
one
r
ank
:
quotient
of
c
onse
cutive
terms
E.2450
Consider
the
sequence
u
n
defined
b
y
the
explicit
form
ula
:
u
n
=
1.2
n
n
for
an
y
n
∈
N
∗
.
1
Giv
e
the
simplified
expression
of
u
n
+1
u
n
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8496
sacados/8496
chapExoCorrec/8497
sacados/8497
chapExoCorrec/8483
sacados/8483
chapExoCorrec/5367
sacados/5367
sacados/10639
chapExoCorrec/8494
sacados/8494
chapExoCorrec/2381
sacados/2381
chapExoCorrec/2522
sacados/2522
chapExoCorrec/5491
sacados/5491
sacados/10471
chapExoCorrec/2451
sacados/2451
chapExoCorrec/2450
sacados/2450
2
Sho
w
that
u
n
is
increasing
from
rank
5
.
E.2670
Determine
the
direction
of
v
ariation
of
the
sequence
u
n
defined
b
y
the
relation
:
u
n
=
n
×
(0.4)
n
for
all
n
∈
N
∗
E.8490
W
e
record
the
sequence
u
n
n
∈
N
∗
defined
b
y
the
explicit
form
ula
:
u
n
=
1.2
n
n
2
for
all
n
∈
N
∗
.
1
Establish
the
follo
wing
iden
tit
y
for
an
y
n
∈
N
∗
:
u
n
+1
u
n
−
1
=
n
2
−
10
·
n
−
5
5
·
(
n
+
1)
2
2
Deduce
that
the
sequence
u
n
is
increasing
from
rank
11
.
E.8489
The
sequence
u
n
n
∈
N
is
defined
b
y
:
u
n
=
n
2
n
+1
for
all
n
∈
N
Sho
w
that
u
n
is
strictly
decreasing
from
rank
2
.
13.
Link
b
etwe
en
r
e
curring
formula
and
explicit
formula
E.2383
1
Consider
the
sequence
u
n
n
∈
N
whose
rank
term
n
is
de-
fined
b
y
the
recurrence
relation
:
u
0
=
2
;
u
n
+1
=
1
3
·
u
n
+
1
for
an
y
n
∈
N
Calculate
the
first
4
terms
of
the
sequence
u
n
.
2
Consider
the
sequence
v
n
n
∈
N
defined
explicitly
b
y
:
v
n
=
1
2
·
1
3
n
+
3
2
a
Calculate
the
first
4
terms
of
the
sequence
v
n
b
Establish
that
for
an
y
natural
n
um
b
er
n
,
w
e
ha
v
e
:
v
n
+1
−
1
3
·
v
n
=
1
3
Deduce
the
equalit
y
of
the
sequences
u
n
and
v
n
.
E.2384
1
Let
u
n
n
∈
N
∗
,
the
sequence
defined
b
y
the
recurrence
re-
lation
and
the
initial
condition
:
u
1
=
1
;
u
n
+1
=
1
1
+
1
u
n
for
an
y
n
∈
N
∗
Calculate
the
first
4
terms
of
the
sequence.
2
Let
v
n
n
∈
N
∗
b
e
the
sequence
whose
term
of
rank
n
has
the
v
alue
:
v
n
=
1
n
for
all
n
∈
N
∗
a
Giv
e
the
v
alue
of
v
1
.
b
Establish
the
follo
wing
iden
tit
y
for
an
y
n
∈
N
∗
:
1
+
1
v
n
=
n
+
1
c
Deduce
that
the
sequence
v
n
follo
ws
the
recurrence
relation
b
elo
w
for
an
y
non-zero
natural
n
um
b
er
:
v
n
+1
=
1
1+
1
v
n
3
What
can
y
ou
sa
y
ab
out
the
sequences
u
n
and
v
n
?
E.5136
Consider
the
t
w
o
sequences
u
n
and
v
n
defined
on
N
b
y
:
u
0
=
0
;
u
n
+1
=
3
·
u
n
−
2
·
n
+
3
for
all
n
∈
N
v
n
=
3
n
+
n
−
1
1
a
Establish
that
the
t
w
o
suites
u
n
and
v
n
ha
v
e
the
same
first
term.
b
Sho
w
that
the
terms
of
the
sequence
v
n
v
erify
the
recurrence
relation
:
v
n
+1
=
3
·
v
n
−
2
·
n
+
3
2
Justify
the
equalit
y
of
the
t
w
o
sequences
u
n
and
v
n
.
E.2385
1
The
sequence
u
n
n
∈
N
is
defined
b
y
the
follo
wing
recur-
rence
relation
:
u
0
=
2
;
u
n
+1
=
2
·
u
n
−
3
·
n
+
2
for
all
n
∈
N
a
By
studying
the
previous
recurrence
relation,
sho
w
that
u
1
=
6
.
b
Determine
the
v
alue
of
the
terms
u
2
and
u
3
.
2
The
terms
of
the
sequence
v
n
n
∈
N
are
defined
b
y
the
relation
:
v
n
=
2
n
+
3
n
+
1
for
all
n
∈
N
a
Giv
e
the
first
4
terms
of
the
sequence
v
n
.
b
F
or
an
y
natural
n
um
b
er
n
(
n
∈
N
)
,
establish
the
equal-
it
y
:
2
·
v
n
−
3
·
n
+
2
=
2
n
+1
+
3
·
n
+
1
+
1
3
Deduce
the
equalit
y
of
the
sequences
u
n
and
v
n
.
14.
Notions
of
limits
:
explicitly
define
d
se
quenc
es
E.8502
Consider
the
sequence
u
n
whose
terms
are
defined
for
an
y
natural
n
um
b
er
n
b
y
the
relation
:
u
n
=
10
·
n
−
1
5
·
n
+1
1
a
Using
the
calculator,
complete
the
table
of
v
alues
b
elo
w
with
v
alues
rounded
to
the
nearest
h
undredth
:
n
0
1
2
3
4
5
u
n
b
In
the
reference
frame
b
elo
w
and
for
n
,
place
the
se-
h
ttps:/
/c
hingmath.fr
chapExoCorrec/2670
sacados/2670
chapExoCorrec/8490
sacados/8490
chapExoCorrec/8489
sacados/8489
chapExoCorrec/2383
sacados/2383
chapExoCorrec/2384
sacados/2384
chapExoCorrec/5136
sacados/5136
chapExoCorrec/2385
sacados/2385
chapExoCorrec/8502
sacados/8502
2
3
4
5
6
I
-1
2
J
O
quence
of
p
oin
ts
A
n
whose
co
ordinates
are
defined
b
y
:
A
n
(
n
;
u
n
)
2
a
Using
the
calculator,
observ
e
the
represen
tativ
e
curv
e
C
f
in
an
orthonormal
frame
of
reference
of
the
function
f
defined
b
y
:
f
(
x
)
=
10
·
x
−
1
5
·
x
+
1
b
What
conjecture
can
b
e
made
ab
out
the
limit
of
the
sequence
u
n
when
n
tends
to
+
∞
?
E.8504
Using
the
calculator,
conjecture
the
limit
of
eac
h
of
the
sequences
defined
b
elo
w
when
n
tends
to
+
∞
:
the
sequence
u
n
defined
for
an
y
natural
n
um
b
er
n
(
n
∈
N
)
b
y
:
u
n
=
4
·
n
1+12
·
n
the
sequence
v
n
defined
for
an
y
natural
n
um
b
er
n
(
n
∈
N
)
b
y
:
v
n
=
n
2
+2
·
n
−
3
the
sequence
w
n
defined
for
an
y
natural
n
um
b
er
n
(
n
∈
N
)
b
y
:
w
n
=
(
−
1)
n
n
+1
E.8503
1
Let
u
n
b
e
an
arithmetic
sequence
with
first
term
u
0
and
reason
r
.
Complete
the
double-en
try
table
b
elo
w,
indicating
the
v
alue
of
lim
n
↦→
+
∞
u
n
in
eac
h
case
r
>
0
r
<
0
u
0
>
0
u
0
<
0
2
Let
v
n
b
e
a
geometric
sequence
with
first
term
u
0
and
reason
q
.
Complete
the
double-en
try
table
b
elo
w,
indi-
cating,
in
eac
h
case
if
p
ossible,
the
v
alue
of
lim
n
↦→
+
∞
v
n
v
0
>
0
v
0
<
0
q
>
1
0
<
q
<
1
−
1
<
q
<
0
q
<
−
1
15.
Notions
of
limits
:
sum
of
the
terms
of
a
se
quenc
e
E.6029
Consider
the
sequence
u
n
geomet-
ric
of
first
term
5
and
reason
2
3
.
Let
S
n
b
e
the
sum
of
the
(
n
+
1)
first
terms
of
the
sequence
u
n
:
S
n
=
u
0
+
u
1
+
·
·
·
+
u
n
−
1
+
u
n
1
Justify
that
the
sequence
S
n
is
increasing.
2
Giv
e
the
expression
of
the
term
S
n
as
a
function
of
n
.
3
a
Using
a
calculator,
complete
the
table
b
elo
w,
round-
ing
v
alues
to
the
nearest
thousandth
:
n
0
1
2
10
20
24
S
n
b
What
conjecture
can
b
e
made
ab
out
the
limit
of
the
sequence
S
n
when
n
tends
to
+
∞
?
E.6013
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
F
or
an
y
natural
n
um
b
er
n
,
let
S
n
b
e
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
:
h
ttps:/
/c
hingmath.fr
2
3
4
5
6
I
-1
2
J
O
chapExoCorrec/8504
sacados/8504
chapExoCorrec/8503
sacados/8503
chapExoCorrec/6029
sacados/6029
chapExoCorrec/6013
sacados/6013
A
B
Etape n
o
0
A
B
Etape n
o
1
A
B
Etape n
o
2
A
B
Etape n
o
3
A
B
Etape n
o
4
A
B
Etape n
o
5
Première ∏gure
5
cm
:::
Seconde ∏gure
3
cm
n
10
100
500
750
1000
u
n
c
What
conjecture
can
b
e
made
ab
out
the
limit
of
the
terms
of
the
sequence
S
n
when
n
tends
to
+
∞
?
E.6014
The
Heige
V
on
K
o
c
h
flak
e
is
con-
structed
as
follo
ws
:
W
e
start
from
a
segmen
t
[
AB
]
of
length
9
cm
.
T
o
mo
v
e
from
one
step
to
the
next,
w
e
cut
eac
h
segmen
t
of
the
figure
in
to
three
equal
parts
and
replace
the
cen
tral
ˇ
ı
segmen
t
with
an
equilateral
triangle.
Here’s
a
represen
tation
of
the
first
6
steps
of
this
construc-
tion
:
A
t
eac
h
n
step,
w
e
note
u
n
the
length
of
the
ˇ
line
brisée
ı
th
us
obtained.
W
e
th
us
construct
a
sequence
of
n
um
b
ers
u
n
defined
for
an
y
natural
n
um
b
er
n
.
1
Determine
the
measure
of
the
first
three
terms
of
the
sequence
u
n
.
2
a
In
step
n
,
express
the
n
um
b
er
of
segmen
ts
s
n
form-
ing
the
ˇ
line
brisée
ı
as
a
function
of
n
.
b
In
step
n
,
express
the
length
‘
n
of
eac
h
of
the
segmen
ts
forming
the
ˇ
line
brisée
ı
as
a
function
of
n
.
3
W
e
note
L
n
the
length
of
the
ˇ
line
brisée
ı
at
step
n
.
This
giv
es
a
sequence
L
n
of
n
umerical
terms
defined
for
an
y
natural
n
um
b
er
n
.
a
Express
the
terms
of
the
sequence
L
n
in
terms
of
their
rank
n
.
b
Complete
the
table
b
elo
w,
rounding
v
alues
to
the
near-
est
h
undredth
of
a
cen
timeter
:
n
0
1
10
20
30
L
n
E.6039
In
this
exercise,
an
y
trace
of
researc
h,
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.
The
qualit
y
of
the
justifications
will
also
b
e
tak
en
in
to
accoun
t.
Consider
the
t
w
o
figures
b
elo
w
:
The
first
figure
is
a
square
whose
sides
measure
5
cm
;
The
second
figure
is
comp
osed
of
a
square
whose
sides
measure
3
cm
,
to
complete
the
figure,
w
e
add
a
new
square
whose
dimensions
ha
v
e
b
een
reduced
b
y
the
co-
efficien
t
4
5
.
W
e
rep
eat
this
figure
a
n
um
b
er
of
times,
but
w
e
don’t
kno
w
ho
w
man
y
times!
Whic
h
of
these
t
w
o
figures
has
the
larger
area?
16.
F
urther
study
:
arithmetic-ge
ometric
se
quenc
es
and
variation
E.9844
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
Establish,
for
an
y
n
∈
N
,
the
equalit
y
:
v
n
+1
=
2
·
v
n
b
Giv
e
the
nature
and
c
haracteristic
elemen
ts
of
the
straigh
t
line
v
n
.
c
Giv
e
the
direction
of
v
ariation
of
the
sequence
v
on
N
.
Justify
y
our
answ
er.
2
a
F
or
an
y
n
∈
N
,
set
:
u
n
+1
−
u
n
=
v
n
+1
−
v
n
b
Deduce
the
direction
of
v
ariations
of
the
sequence
u
n
on
N
.
E.9845
Consider
the
sequence
u
n
)
defined
on
N
∗
b
y
:
u
1
=
0
;
u
n
+1
=
0.2
·
u
n
+
0.04
for
an
y
n
∈
N
∗
1
W
e
define
the
sequence
v
n
on
N
∗
b
y
the
relation
:
v
n
=
u
n
−
0.05
a
Establish
equalit
y
for
an
y
n
∈
N
∗
:
v
n
+1
=
0.2
·
v
n
b
Determine
the
nature
and
c
haracteristic
elemen
ts
of
the
sequence
v
n
.
2
a
Establish
equalit
y
for
an
y
n
∈
N
∗
:
u
n
+1
−
u
n
=
v
n
+1
−
v
n
b
Deduce
the
direction
of
v
ariation
of
the
sequence
u
n
E.6668
Consider
the
sequence
u
n
defined
b
y
:
u
0
=
2
;
u
n
+1
=
1
+
2
·
u
n
−
6
for
all
n
∈
N
and
the
sequence
v
n
,
defined
for
an
y
n
∈
N
,
b
y
:
v
n
=
u
n
−
3
1
Establish
that
the
sequence
v
n
is
a
geometric
sequence
whose
c
haracteristic
elemen
ts
and
direction
of
v
ariations
will
b
e
giv
en.
2
a
F
or
an
y
n
∈
N
,
establish
equalit
y
:
u
n
+1
−
u
n
=
v
n
+1
−
v
n
b
Determine
the
direction
of
v
ariation
of
the
sequence
u
n
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6014
sacados/6014
A
B
Etape n
o
0
A
B
Etape n
o
1
A
B
Etape n
o
2
A
B
Etape n
o
3
A
B
Etape n
o
4
A
B
Etape n
o
5
chapExoCorrec/6039
sacados/6039
Première ∏gure
5
cm
:::
Seconde ∏gure
3
cm
chapExoCorrec/9844
sacados/9844
chapExoCorrec/9845
sacados/9845
chapExoCorrec/6668
sacados/6668
17.
F
urther
study
:
arithmetic-ge
ometric
se
quenc
es
and
explicit
formulas
E.9843
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
+
∞
.
E.9842
W
e
wish
to
study
the
sequence
(
u
n
)
of
first
term
u
0
=
5
defined
b
y
the
follo
wing
recurrence
rela-
tion
:
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
a
Sho
w
that
the
sequence
(
v
n
)
is
a
geometric
sequence
whose
first
term
and
reason
will
b
e
sp
ecified.
b
Express
v
n
as
a
function
of
rank
n
.
2
a
Deduce
the
expression
of
u
n
as
a
function
of
n
.
b
F
or
an
y
n
∈
N
,
establish
that
:
u
n
+1
−
u
n
=
2
3
·
1
3
n
c
Deduce
the
direction
of
v
ariations
of
the
sequence
u
n
.
E.5368
Consider
the
sequence
u
n
defined
b
y
:
u
0
=
3
;
u
n
+1
=
3
4
·
u
n
+
1
2
for
an
y
in
teger
n
∈
N
1
Consider
the
sequence
v
n
defined
b
y
the
follo
wing
re-
lation
for
an
y
natural
n
um
b
er
n
:
v
n
=
1
2
·
u
n
−
1
a
Establish
the
equalit
y
b
elo
w
for
an
y
natural
n
um
b
er
n
:
v
n
+1
=
3
4
·
v
n
b
Giv
e
the
direction
of
v
ariation
of
the
sequence
v
n
.
2
Deduce
the
direction
of
v
ariation
of
the
sequence
u
n
.
18.
Mor
e
:
other
uses
for
auxiliary
suites
E.8487
Consider
the
sequence
v
n
defined
b
y
:
v
0
=
1
;
v
n
+1
=
v
n
+
2
·
n
+
3
for
all
n
∈
N
1
Determine
the
first
fiv
e
terms
of
the
sequence
v
n
.
2
What
conjecture
can
b
e
made
ab
out
the
terms
of
the
sequence
v
n
?
W
e
define
the
sequence
w
n
defined
b
y
:
w
n
=
v
n
+1
−
v
n
for
an
y
n
∈
N
1
Justify
that
the
sequence
w
n
is
an
arithmetic
sequence.
Sp
ecify
the
c
haracteristic
elemen
ts
of
this
sequence.
2
Determine
the
expression
of
the
sum
S
of
the
first
n
terms
of
the
sequence
w
n
.
3
Noting
the
equalit
y
n
−
1
k
=0
w
k
+
v
0
=
v
n
,
deduce
the
expres-
sion
of
the
term
v
n
as
a
function
of
n
.
4
Confirm
the
conjecture
made
in
question
b
.
19.
F
urther
development
:
jointly
define
d
suites
E.8498
Consider
the
t
w
o
sequences
u
n
and
v
n
defined
for
an
y
natural
n
um
b
er
n
b
y
the
relations
:
u
0
=
3
v
0
=
1
;
u
n
+1
=
2
·
u
n
+
v
n
2
v
n
+1
=
u
n
+
3
·
v
n
3
1
Determine
the
first
three
terms
of
the
sequences
u
n
and
v
n
.
2
A
dmit
that
the
terms
of
the
sequences
u
n
and
v
n
are
strictly
p
ositiv
e
n
um
b
ers.
Sho
w
that
the
sequences
u
n
and
v
n
are
increasing
on
N
.
E.4235
Consider
the
sequences
u
n
and
v
n
defined
b
y
:
u
0
=
0
;
u
n
+1
=
u
n
+
v
n
2
for
all
n
∈
N
v
0
=
12
;
v
n
+1
=
u
n
+
2
·
v
n
3
for
an
y
n
∈
N
1
Demonstrate
that
the
sequence
w
n
defined
b
y
w
n
=
v
n
−
u
n
is
a
con
v
ergen
t
geometric
sequence
and
that
all
its
terms
are
p
ositiv
e.
2
Sho
w
that
the
sequence
u
n
is
increasing
and
then
that
the
sequence
v
n
is
decreasing.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/9843
sacados/9843
chapExoCorrec/9842
sacados/9842
chapExoCorrec/5368
sacados/5368
chapExoCorrec/8487
sacados/8487
chapExoCorrec/8498
sacados/8498
chapExoCorrec/4235
sacados/4235
Motif
0
Motif
1
Motif
2
Motif
3
Motif
0
Motif
1
Motif
2
Motif
3
20.
Unclassifie
d
financial
ye
ars
E.5973
1
a
Consider
the
sequence
u
n
defined
b
y
the
relation
:
u
0
=
0
;
u
n
+1
=
1
2
−
u
n
for
all
n
∈
N
Determine
the
first
four
terms
of
the
sequence
u
n
.
b
Consider
the
sequence
v
n
defined
b
y
the
relation
:
v
n
=
n
n
+
1
for
an
y
n
∈
N
.
Determine
the
first
four
terms
of
the
sequence
v
n
.
c
What
conjecture
can
b
e
made
ab
out
the
sequences
u
n
and
v
n
?
2
a
Simplify
the
follo
wing
expression
:
v
n
+1
·
2
−
v
n
b
Justify
that
the
t
w
o
suites
u
n
and
v
n
are
equal.
E.10569
Consider
the
step-b
y-step
pattern
con-
struction
whose
first
fiv
e
steps
are
sho
wn
b
elo
w
:
1
Ho
w
man
y
tiles
are
needed
to
construct
pattern
4?
pat-
tern
5?
2
W
e
note
u
n
the
n
um
b
er
of
tiles
required
to
construct
pattern
n
.
a
Giv
e
the
recurrence
form
ula
v
erified
b
y
the
sequence
u
n
.
b
Establish
that
the
terms
of
the
sequence
u
n
admit
the
explicit
form
ula
:
u
n
=
2
n
2
+
5
n
+
4
E.10570
Consider
the
step-b
y-step
pattern
con-
struction
whose
first
fiv
e
steps
are
sho
wn
b
elo
w
:
1
Ho
w
man
y
tiles
are
needed
to
construct
pattern
4?
pat-
tern
5?
2
W
e
note
u
n
the
n
um
b
er
of
tiles
needed
to
construct
pat-
tern
n
.
a
Giv
e
the
recurrence
form
ula
v
erified
b
y
the
sequence
u
n
.
b
Establish
that
the
terms
of
the
sequence
u
n
v
erify
the
explicit
form
ula
:
u
n
=
1
3
·
n
3
+
3
2
·
n
2
+
13
6
·
n
+
1
h
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sacados/10569
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