a
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4
3
2
1
Outside the high school program
/ Sequences
39 exercises (including 38 corrected)
a
Any sequences
(14 exercices)
Any sequences and graphical reading
(5 exercices)
Sum of terms
(10 exercices)
Course - old program: Suites
(3 exercices)
Recalls on suites
(3 exercices)
1
er
étap
e
2
ième
étap
e
3
ième
étap
e
1
er
étape
2
e
étape
3
e
étape
1.
A
ny
se
quenc
es
E.7716
F
or
eac
h
question,
determine
the
first
four
terms
of
the
sequence
u
n
n
∈
N
:
a
u
n
=
n
+
1
n
+
2
b
u
n
+1
=
2
·
u
n
−
2
;
u
0
=
3
c
u
n
=
n
2
+
n
+
1
d
u
n
+1
=
2
·
u
n
−
2
;
u
0
=
1
e
u
n
=
n
−
2
n
+
1
f
u
n
+1
=
u
n
−
2
u
n
+
1
;
u
0
=
2
E.4556
Consider
the
construction
of
a
house
of
cards
:
F
or
an
y
non-zero
natural
n
um
b
er
(
n
∈
N
∗
)
,
w
e
note
u
n
the
n
um
b
er
of
cards
needed
to
build
the
n
ème
step.
Th
us,
w
e
ha
v
e
:
u
1
=
2
;
u
2
=
:
:
:
;
u
3
=
:
:
:
;
u
4
=
:
:
:
E.7498
The
figures
b
elo
w
are
constructed
using
small
w
o
o
den
stic
ks.
F
or
n
a
strictly
p
ositiv
e
in
teger
(
n
∈
N
∗
)
,
,
w
e
denote
u
n
the
n
um
b
er
of
stic
ks
needed
for
construction
in
step
n
.
W
e
there-
fore
ha
v
e
:
u
1
=
10
;
u
2
=
:
:
:
;
u
3
=
:
:
:
;
u
4
=
:
:
:
E.7598
Consider
the
sequence
u
n
defined
for
an
y
in
teger
n
p
ositiv
e
or
zero
(
n
∈
N
)
b
y
:
u
0
=
2
;
u
n
+1
=
u
n
+
n
n
+
1
Determine
the
first
four
terms
of
the
sequence
u
n
.
E.7492
Consider
the
follo
wing
sequences
of
n
um-
b
ers
:
a
a
0
=
4
;
a
1
=
7
;
a
2
=
10
;
a
3
=
13
;
a
4
=
16
.
.
.
b
b
0
=
1
;
b
1
=
−
2
;
b
2
=
4
;
b
3
=
−
8
;
b
4
=
16
.
.
.
c
c
0
=
1
;
c
1
=
3
;
c
2
=
5
;
c
3
=
7
;
c
4
=
9
.
.
.
d
d
0
=
16
;
d
1
=
8
;
d
2
=
4
;
d
3
=
2
;
d
4
=
1
.
.
.
Asso
ciate
eac
h
of
these
sequences
with
one
of
the
follo
wing
relations,
whic
h
defines
the
v
alue
of
a
term
either
in
terms
of
the
v
alue
of
its
predecessor
or
in
terms
of
its
rank
:
1
u
n
+1
=
u
n
+
2
2
u
n
=
4
+
3
·
n
3
u
n
+1
=
−
2
·
u
n
4
u
n
=
16
×
1
2
n
E.4557
1
Here
are
some
examples
of
n
um
b
er
sequences
:
a
(
2
;
5
;
8
;
11
;
14
;
:
:
:
)
b
(
2
;
6
;
18
;
54
;
162
;
:
:
:
)
c
(
6
;
−
6
;
6
;
−
6
;
6
;
:
:
:
)
d
(
1
;
3
;
7
;
15
;
31
;
:
:
:
)
F
or
eac
h
of
these
n
um
b
er
sequences,
find
the
relationship
that
allo
ws
y
ou
to
obtain
a
v
alue
based
on
the
previous
v
alues.
2
Here
are
some
other
examples
of
n
um
b
er
sequences
:
a
(
0
;
2
;
4
;
6
;
8
;
:
:
:
)
b
(
1
;
6
;
11
;
16
;
21
;
:
:
:
)
c
(
1
;
2
;
4
;
8
;
16
;
32
;
:
:
:
)
d
(
1
;
2
;
3
;
2
;
√
5
;
√
6
;
:
:
:
)
e
2
;
3
2
;
4
3
;
5
4
;
6
5
;
:
:
:
F
or
eac
h
of
these
sequences
of
n
um
b
ers,
find
the
rela-
tionship
that
allo
ws
y
ou
to
obtain
a
v
alue
based
on
its
p
osition
in
the
sequence.
E.4559
Consider
the
sequences
defined
b
elo
w
b
y
the
v
alue
of
their
first
term
and
a
recurrence
relation
where
the
in
teger
n
denotes
a
p
ositiv
e
or
zero
in
teger
(
n
∈
N
)
:
a
u
0
=
5
;
u
n
+1
=
u
n
+
2
b
v
0
=
3
;
v
n
+1
=
2
·
v
n
c
w
0
=
2
;
w
n
+1
=
−
w
n
d
x
0
=
4
;
x
n
+1
=
2
·
x
n
−
2
d
y
0
=
1
;
y
1
=
1
;
y
n
+1
=
y
n
+
y
n
−
1
Determine
the
first
fiv
e
terms
of
eac
h
of
these
sequences.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7716
sacados/7716
chapExoCorrec/4556
sacados/4556
1
er
étap
e
2
ième
étap
e
3
ième
étap
e
chapExoCorrec/7498
sacados/7498
1
er
étape
2
e
étape
3
e
étape
chapExoCorrec/7598
sacados/7598
chapExoCorrec/7492
sacados/7492
chapExoCorrec/4557
sacados/4557
chapExoCorrec/4559
sacados/4559
n
0
10
20
30
40
50
u
n
-2
-1
1
2
3
4
-4
-3
-2
-1
2
3
4
I
-3
-2
-1
2
3
4
J
O
C
f
E.4581
F
or
eac
h
question,
a
sequence
is
defined
where
the
rank
n
denotes
a
p
ositiv
e
in
teger
or
zero
(
n
∈
N
)
.
Determine
the
first
fiv
e
terms
of
the
sequence
u
n
:
a
u
n
=
n
+
1
n
+
2
b
u
n
+1
=
2
·
u
n
−
2
;
u
0
=
3
c
u
n
=
n
2
+
n
+
1
d
u
n
+1
=
2
·
u
n
−
2
;
u
0
=
1
e
u
n
=
n
−
2
n
+
1
f
u
n
+1
=
u
n
−
2
u
n
+
1
;
u
0
=
2
E.4604
1
Consider
the
sequence
u
n
whose
rank
term
n
,
a
p
osi-
tiv
e
or
zero
in
teger,
is
giv
en
b
y
the
relation
:
u
n
=
−
7
4
·
1
−
(
−
1)
n
+
3
a
Determine
the
first
fiv
e
terms
of
this
sequence.
b
What
can
b
e
said
ab
out
the
v
alue
of
the
terms
in
the
sequence
u
n
?
2
Consider
the
sequence
v
n
defined,
for
an
y
p
ositiv
e
or
zero
rank,
b
y
:
v
n
+1
=
1
−
v
n
1
+
v
n
;
v
0
=
3
a
Determine
the
first
fiv
e
terms
of
the
sequence
v
n
.
b
What
do
w
e
notice?
E.7539
Consider
the
t
w
o
algorithms
b
elo
w
:
Algorithm
1
u
←
4
For
i
ranging
from
1
to
53
u
←
u
+
3
End
For
Algorithm
2
u
←
1
For
i
ranging
from
1
to
4
u
←
2
×
u
+
1
End
For
F
or
eac
h
algorithm,
giv
e
the
v
alue
con
tained
in
v
ariable
u
after
the
algorithm
has
b
een
executed.
E.7591
1
Determine
the
first
four
terms
of
the
sequence
u
n
de-
fined
for
an
y
p
ositiv
e
or
zero
in
teger
(
n
∈
N
)
:
u
0
=
3
;
u
n
+1
=
2
·
u
n
−
2
2
Determine
the
first
four
terms
of
the
sequence
v
n
de-
fined
for
an
y
p
ositiv
e
or
zero
in
teger
(
n
∈
N
)
:
v
n
=
n
+
1
2
·
n
+
1
E.7597
Consider
the
sequence
u
n
defined
for
an
y
in
teger
n
p
ositiv
e
or
zero
(
n
∈
N
)
b
y
:
u
0
=
3
;
u
n
+1
=
2
·
u
n
+
1
n
+
1
Determine
the
first
four
terms
of
the
sequence
u
n
.
E.7592
1
Determine
the
first
four
terms
of
the
sequence
u
n
de-
fined
for
an
y
p
ositiv
e
or
zero
in
teger
(
n
∈
N
)
:
u
0
=
4
;
u
n
+1
=
3
−
2
·
u
n
2
Determine
the
first
four
terms
of
the
sequence
v
n
de-
fined
for
an
y
p
ositiv
e
or
zero
in
teger
(
n
∈
N
)
:
v
n
=
2
·
n
−
1
n
+
3
E.4558
Consider
the
n
umerical
sequences
de-
fined
b
elo
w,
where
their
rank
n
is
a
p
ositiv
e
in
teger
or
zero
(
n
∈
N
)
:
a
u
n
=
2
n
b
v
n
=
3
n
−
4
c
w
n
=
n
2
+
3
d
x
n
=
2
n
Determine
the
first
fiv
e
terms
of
eac
h
of
these
sequences.
2.
A
ny
se
quenc
es
and
gr
aphic
al
r
e
ading
E.7750
Consider
a
sequence
u
n
defined
for
an
y
natural
n
um
b
er
n
p
ositiv
e
or
zero
(
n
∈
N
)
.
In
a
reference
frame
are
represen
ted
the
p
oin
ts
with
co
ordinates
(
n
;
u
n
)
for
n
b
et
w
een
0
and
50
:
1
Giv
e
the
exact
v
alues
of
u
0
and
u
10
.
2
Giv
e
appro
ximate
v
alues
of
u
5
,
u
30
and
u
40
.
3
What
can
b
e
said
ab
out
the
v
alues
of
the
terms
u
n
as
the
v
alue
of
n
increases?
E.4584
In
the
plane
pro
vided
with
an
orthonor-
mal
reference
frame
O
;
I
;
J
,
consider
the
represen
tation
C
f
of
a
function
f
defined
on
the
in
terv
al
[
−
4
;
4]
:
Consider
the
sequences
(
u
n
)
n
∈
N
and
(
v
n
)
n
∈
N
v
erifying
the
re-
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4581
sacados/4581
chapExoCorrec/4604
sacados/4604
chapExoCorrec/7539
sacados/7539
chapExoCorrec/7591
sacados/7591
chapExoCorrec/7597
sacados/7597
chapExoCorrec/7592
sacados/7592
chapExoCorrec/4558
sacados/4558
chapExoCorrec/7750
sacados/7750
n
0
10
20
30
40
50
u
n
-2
-1
1
2
3
4
chapExoCorrec/4584
sacados/4584
-4
-3
-2
-1
2
3
4
I
-3
-2
-1
2
3
4
J
O
C
f
-4
-3
-2
-1
2
3
4
I
-4
-3
-2
-1
2
3
4
J
O
C
f
-2
-1
2
3
4
5
6
I
-1
2
3
4
5
6
7
J
O
C
f
-1
2
3
4
5
6
7
8
I
-1
2
3
4
J
O
C
f
lations
:
u
n
+1
=
f
u
n
;
v
n
+1
=
f
v
n
v
erifying
the
follo
wing
initial
conditions
:
u
0
=
−
1
;
v
0
=
−
4
Determine
the
first
100
terms
of
eac
h
of
these
t
w
o
sequences.
E.4583
Consider
the
function
f
defined
on
[
−
4
;
4]
don
t
represen
tativ
e
curv
e
C
f
est
giv
en
b
elo
w
:
Consider
the
sequence
u
n
n
∈
N
defined
b
y
the
relation
u
0
=
2
and
u
n
+1
=
f
u
n
.
1
Sho
w
that
the
term
u
1
est
equals
−
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
E.4610
In
the
plane
pro
vided
with
a
O
;
I
;
J
orthonormal,
consider
the
curv
e
C
f
represen
tativ
e
of
the
func-
tion
f
defined
b
y
the
relation
:
f
(
x
)
=
−
12
x
+
36
x
−
24
The
line
(Δ)
is
the
first
bisector
of
the
plane.
Consider
the
sequence
u
n
n
∈
N
defined
b
y
recurrence
b
y
:
u
n
+1
=
f
u
n
;
u
0
=
−
2
1
Graphically
,
place
on
the
x-axis
the
first
fiv
e
v
alues
of
the
sequence
u
n
.
2
Determine,
b
y
calculation,
the
v
alue
of
the
first
three
terms
of
the
sequence
u
n
.
E.4609
In
the
plane
pro
vided
with
a
O
;
I
;
J
orthonormal,
consider
the
curv
e
C
f
of
the
function
f
whose
image
of
a
n
um
b
er
x
is
defined
b
y
:
f
(
x
)
=
−
x
+
4
x
+
2
The
line
(Δ)
is
the
first
bisector
of
the
plane.
Consider
the
sequence
u
n
n
∈
N
defined
b
y
the
relation
:
u
n
+1
=
f
u
n
;
u
0
=
8
1
On
the
x-axis,
sho
w
the
v
alues
of
the
first
six
terms
of
the
sequence.
2
Determine,
b
y
calculation,
the
first
three
terms
of
the
sequence
u
n
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4583
sacados/4583
-4
-3
-2
-1
2
3
4
I
-4
-3
-2
-1
2
3
4
J
O
C
f
chapExoCorrec/4610
sacados/4610
-2
-1
2
3
4
5
6
I
-1
2
3
4
5
6
7
J
O
C
f
chapExoCorrec/4609
sacados/4609
-1
2
3
4
5
6
7
8
I
-1
2
3
4
J
O
C
f
3.
Sum
of
terms
E.7724
1
Consider
the
arithmetic
sequence
u
n
with
first
term
4
and
reason
2
3
a
Determine
the
first
three
terms
of
the
sequence
u
n
.
b
Determine
the
rank
of
the
sequence
u
n
ha
ving
v
alue
38
.
c
Determine
the
sum
of
terms
:
S
=
u
0
+
u
1
+
·
·
·
+
u
99
.
2
Consider
the
geometric
sequence
v
n
of
first
term
4
and
reason
2
3
a
Determine
the
first
three
terms
of
the
sequence
v
n
.
b
Determine
the
sum
of
the
terms
:
S
=
v
0
+
v
1
+
·
·
·
+
v
15
.
E.7798
1
Consider
the
arithmetic
sequence
u
n
with
first
term
1
3
and
reason
1
2
a
Determine
the
first
three
terms
of
the
sequence
u
n
.
b
Determine
the
rank
of
the
term
u
n
ha
ving
v
alue
77
6
.
c
Determine
the
sum
of
terms
:
S
=
u
0
+
u
1
+
·
·
·
+
u
99
.
2
Consider
the
geometric
sequence
v
n
of
first
term
1
3
and
reason
1
2
a
Determine
the
first
three
terms
of
the
sequence
v
n
.
b
Determine
the
sum
of
the
terms
:
S
=
v
0
+
v
1
+
·
·
·
+
v
15
.
E.6547
1
Consider
the
suite
u
n
n
∈
N
arithmetic
with
first
term
5
and
reason
3
.
Determine
the
v
alue
of
the
sum
S
of
the
100
first
terms
of
this
sequence.
2
Consider
the
geometric
sequence
v
n
n
∈
N
of
first
term
3
and
reason
1
4
.
Determine
the
v
alue
of
the
sum
S
of
the
first
100
terms
of
this
sequence.
E.2453
1
Let
u
n
)
n
∈
N
b
e
an
arithmetic
sequence
of
first
term
2
and
reason
3
5
,
determine
the
v
alue
of
the
follo
wing
sum
:
S
=
u
5
+
u
6
+
·
·
·
+
u
21
2
Consider
the
follo
wing
sum
whose
terms
are
those
of
a
geometric
sequence
:
S
=
16
+
24
+
36
+
·
·
·
+
3
10
2
6
Determine
the
v
alue
of
the
sum
S
.
E.5172
Consider
the
sequence
u
n
defined
b
y
:
u
0
=
0
;
u
n
+1
=
−
1
2
·
u
n
+
1
for
all
n
∈
N
1
Determine
the
first
fiv
e
terms
of
the
sequence
u
n
.
2
Consider
the
sequence
v
n
defined
b
y
:
v
n
=
u
n
−
2
3
for
all
n
∈
N
a
Determine
the
first
four
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
2
·
v
n
c
Giv
e
the
nature
and
v
alues
of
the
c
haracteristic
ele-
men
ts
of
the
sequence
v
n
.
3
a
Determine
the
v
alue
of
the
sum
S
defined
b
y
:
S
=
v
0
+
v
1
+
·
·
·
+
v
14
b
Determine
the
v
alue
of
the
sum
S
defined
b
y
:
S
=
u
0
+
u
1
+
·
·
·
+
u
14
4
a
Giv
e
the
expression
of
the
term
v
n
as
a
function
of
its
rank
n
.
b
Giv
e
the
expression
of
the
term
u
n
as
a
function
of
its
rank
n
.
E.3017
1
Consider
a
sequence
u
n
n
∈
N
arithmetic
suc
h
that
:
u
13
=
7
;
u
20
=
35
2
a
Justifying
y
our
approac
h,
find
the
c
haracteristic
ele-
men
ts
of
this
sequence.
b
Deduce
the
v
alue
of
the
sum
S
defined
b
y
:
S
=
u
5
+
u
6
+
·
·
·
+
u
22
2
Consider
a
geometric
sequence
v
n
n
∈
N
suc
h
that
:
v
9
=
5
6
7
4
;
v
16
=
5
13
7
11
a
Justifying
y
our
approac
h,
find
the
c
haracteristic
ele-
men
ts
of
this
sequence.
b
Deduce
the
v
alue
of
the
sum
S
defined
b
y
:
S
=
v
5
+
v
6
+
·
·
·
+
v
22
E.7607
1
Consider
the
sequence
u
n
geometric
with
first
term
2
and
reason
3
.
Determine
the
sum
of
the
first
10
terms
of
the
sequence
u
n
.
2
Consider
the
sequence
v
n
geometric
of
first
term
5
and
reason
1
2
.
Determine
an
expression
for
the
sum
S
defined
b
y
:
S
=
v
0
+
v
1
+
v
2
+
·
·
·
+
v
12
Then
giv
e
the
v
alue
of
S
rounded
to
the
nearest
h
un-
dredth.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7724
sacados/7724
Merci Tofeil! 1s2
chapExoCorrec/7798
sacados/7798
chapExoCorrec/6547
sacados/6547
chapExoCorrec/2453
sacados/2453
chapExoCorrec/5172
sacados/5172
chapExoCorrec/3017
sacados/3017
chapExoCorrec/7607
sacados/7607
1
er
étap
e
2
ième
étap
e
3
ième
étap
e
E.8471
The
terms
of
eac
h
sum
are
the
terms
of
a
geometric
sequence.
Determine
the
v
alue
of
these
t
w
o
sums
:
1
S
1
=
1
+
1
2
+
1
4
+
1
8
+
·
·
·
+
1
1024
2
S
2
=
2
+
3
+
3
2
+
3
3
4
+
9
8
E.8467
Consider
the
sum
S
defined
b
y
:
S
=
5
+
2
−
1
−
4
−
·
·
·
−
37
Deduce
the
v
alue
of
S
.
E.7663
Consider
the
sum
S
defined
b
y
:
S
=
1
+
5
2
+
4
+
11
2
+
·
·
·
+
100
W
e
admit
that
the
terms
of
the
sum
S
are
the
first
successiv
e
terms
of
a
sequence
u
n
arithmetic
defined
on
N
.
1
Donner
les
élémen
ts
caractéristiques
de
la
suite
u
n
et
déterminer
le
rang
du
terme
de
la
suite
a
y
an
t
100
p
our
v
aleur.
2
Deduce
the
v
alue
of
the
sum
S
.
4.
Course
-
old
pr
o
gr
am
:
Suites
E.3291
The
follo
wing
results
are
assumed
to
b
e
kno
wn
:
Definition-prop
osition
:
(1)
t
w
o
sequences
(
u
n
)
and
(
v
n
)
are
said
to
b
e
adjacen
t
when
:
one
is
increasing,
the
other
is
decreasing,
and
u
n
−
v
n
tends
to
w
ard
0
as
n
tends
to
w
ard
+
∞
;
(2)
if
(
u
n
)
and
(
v
n
)
are
t
w
o
adjacen
t
sequences
suc
h
that
(
u
n
)
is
increasing
and
(
v
n
)
is
decreasing,
then
for
an
y
n
b
elonging
to
N
,
w
e
ha
v
e
:
u
n
v
n
;
(3)
an
y
increasing
and
b
ounded
sequence
is
con
v
ergen
t
;
an
y
decreasing
and
b
ounded
sequence
is
con
v
ergen
t.
Then
pro
v
e
the
follo
wing
prop
osition
:
ˇ
T
w
o
adjacen
t
sequences
are
con
v
ergen
t
and
ha
v
e
the
same
limit
ı.
E.3445
Consider
the
three
sequences
u
n
n
∈
N
,
v
n
n
∈
N
,
w
n
n
∈
N
whic
h
v
erify
the
follo
wing
conditions
:
The
t
w
o
sequences
v
n
and
w
n
con
v
erge
to
a
real
n
um-
b
er
‘
.
F
rom
a
rank
n
0
and
for
an
y
in
teger
n
greater
than
n
0
,
w
e
ha
v
e
:
v
C
O
P
Y
06
u
n
w
n
Sho
w
that
the
sequence
u
n
is
con
v
ergen
t
and,
more
pre-
cisely
,
has
the
follo
wing
limit
:
lim
n
↦→
+
∞
u
n
=
‘
E.3475
Using
the
definition
and
the
t
w
o
prop
erties
b
elo
w,
sho
w
that
if
u
n
and
v
n
are
t
w
o
adjacen
t
sequences,
then
they
are
con
v
ergen
t
and
ha
v
e
the
same
limit.
Definition
:
T
w
o
sequences
are
adjacen
t
when
one
is
increasing,
the
other
is
decreasing
and
the
difference
of
the
t
w
o
con-
v
erges
to
0
.
Prop
ert
y
1
:
If
t
w
o
sequences
u
n
and
v
n
are
adjacen
t
with
u
n
increasing
and
v
n
decreasing
then
for
an
y
natural
n
um
b
er
:
u
n
v
n
.
Prop
ert
y
2
:
an
y
increasing
and
ma
joring
sequence
con
v
erges
;
an
y
decreasing
and
minoring
sequence
con
v
erges.
5.
R
e
c
al
ls
on
suites
E.6124
Consider
the
construction
of
a
house
of
cards
:
Ho
w
man
y
cards
are
needed
to
complete
the
4
ième
stage
of
this
construction?
for
the
5
ième
stage?
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8471
sacados/8471
chapExoCorrec/8467
sacados/8467
chapExoCorrec/7663
sacados/7663
chapExoCorrec/3291
sacados/3291
Extrait France Juin 2005
chapExoCorrec/3445
sacados/3445
chapExoCorrec/3475
sacados/3475
chapExoCorrec/6124
sacados/6124
1
er
étap
e
2
ième
étap
e
3
ième
étap
e
1
er
étape
2
ième
étape
3
ième
étape
u
n
−
1
u
n
u
n
u
n
u
n
−
1
u
n
u
n
u
n
u
n
−
1
u
n
u
n
u
n
u
n
−
1
u
n
u
n
u
n
0,2
0,4
0,6
0,8
I
0,2
0,4
0,6
0,8
J
O
u
0
u
1
f
(
u
0
E.6144
Consider
the
follo
wing
constructions
:
Note
u
n
the
n
umerical
sequence
defined
on
N
∗
where
u
n
represen
ts
the
n
um
b
er
of
matc
hes
required
to
construct
the
n
ième
step.
Whic
h
of
the
follo
wing
definitions
can
b
e
used
to
define
the
sequence
u
n
sho
wn
ab
o
v
e
:
a
u
1
=
4
u
n
+1
=
4
·
u
n
b
u
1
=
4
u
n
+1
=
u
n
+
4
·
n
c
u
1
=
4
u
n
+1
=
u
n
+
4
·
(
n
+
1)
d
u
1
=
4
u
n
+1
=
4
·
u
n
+
n
E.6125
Consider
the
n
umerical
sequences,
b
e-
lo
w,
defined
on
N
b
y
recurrence
:
i.e.
according
to
their
initial
v
alue
and
a
relationship
with
the
preceding
terms.
1
u
0
=
5
;
u
n
+1
=
u
n
+2
for
all
n
∈
N
.
2
v
0
=
3
;
v
n
+1
=
2
·
v
n
for
an
y
n
∈
N
.
3
w
0
=
2
;
w
n
+1
=
−
w
n
for
an
y
n
∈
N
.
4
x
0
=
4
;
x
n
+1
=
2
·
x
n
−
2
for
an
y
n
∈
N
5
x
0
=
1
;
x
1
=
1
;
x
n
+1
=
x
n
+
x
n
−
1
for
all
n
∈
N
Determine
the
first
fiv
e
terms
of
eac
h
of
these
sequences.
6.
Unclassifie
d
financial
ye
ars
E.4644
Consider
the
sequence
u
n
defined
on
N
b
y
the
follo
wing
recurrence
relation
:
u
0
=
0
;
u
n
+1
=
u
n
−
2
n
+
11
1
a
Using
the
soft
w
are
of
y
our
c
hoice,
plot
the
scatter-
plot
asso
ciated
with
the
first
15
terms
of
this
sequence.
b
Mak
e
a
conjecture
as
to
the
nature
of
the
curv
e
passing
through
these
p
oin
ts.
2
a
Determine
the
function
f
defined
b
y
a
p
olynomial
of
the
second
degree,
v
erifying
the
relations
:
u
0
=
f
(0)
;
u
1
=
f
(1)
;
u
11
=
f
(11)
b
Giv
e
the
reduced
expression
of
the
expression
:
f
(
x
+
1)
−
f
(
x
)
.
c
Establish
the
expression
of
the
terms
of
the
sequence
u
n
as
a
function
of
their
rank
n
.
E.5105
Consider
the
sequence
u
n
defined
b
y
:
u
0
=
1
;
u
n
+1
=
−
1
2
·
u
n
+
1
for
all
n
∈
N
1
Determine
the
first
four
terms
of
this
sequence.
2
a
Establish
the
follo
wing
equalit
y
for
all
n
∈
N
:
u
n
+2
=
1
4
·
u
n
+
1
2
b
Establish
the
follo
wing
equalit
y
for
all
n
∈
N
:
u
n
+2
=
u
n
+
u
n
+1
2
3
When
represen
ting
four
terms
of
the
sequence
u
n
on
a
graduated
line,
whic
h
of
the
four
prop
ositions
b
elo
w
w
ould
b
est
represen
t
them?
a
b
c
d
E.6635
Consider
the
function
f
defined
on
0
;
1
b
y
the
relation
:
f
(
x
)
=
5
4
−
1
x
+
1
1
a
Establish
the
follo
wing
v
alues
:
f
1
4
=
9
20
;
f
◦
f
1
4
=
65
116
b
Determine
the
v
alue
of
:
f
◦
f
◦
f
1
4
Belo
w
is
the
represen
tativ
e
curv
e
C
f
of
the
function
f
in
the
orthonormal
co
ordinate
system
O
;
I
;
J
:
2
a
Giv
e
the
v
alues
rounded
to
the
nearest
thousandth
of
the
follo
wing
n
um
b
ers
:
u
0
=
1
4
=
:
:
:
:
:
:
u
1
=
9
20
≈
:
:
:
:
:
:
u
2
=
65
116
≈
:
:
:
:
:
:
u
3
=
441
724
≈
:
:
:
:
:
:
b
Place
the
v
alues
u
2
and
u
3
on
the
x-axis.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6144
sacados/6144
1
er
étape
2
ième
étape
3
ième
étape
chapExoCorrec/6125
sacados/6125
sacados/4644
chapExoCorrec/5105
sacados/5105
u
n
−
1
u
n
u
n
u
n
u
n
−
1
u
n
u
n
u
n
u
n
−
1
u
n
u
n
u
n
u
n
−
1
u
n
u
n
u
n
chapExoCorrec/6635
sacados/6635
0,2
0,4
0,6
0,8
I
0,2
0,4
0,6
0,8
J
O
u
0
u
1
f
(
u
0
c
Place
the
v
alues
f
(
u
1
)
,
f
(
u
2
)
,
and
f
(
u
3
)
on
the
y-axis.
3
a
Dra
w
the
line
segmen
t
connecting
the
t
w
o
p
oin
ts
A
1
(
u
1
;
0
)
and
B
1
(
0
;
f
(
u
0
)
)
.
What
is
the
nature
of
triangle
O
A
1
B
1
?
b
F
or
i
ranging
from
1
to
3
,
w
e
define
the
p
oin
ts
:
A
i
(
u
i
;
0
)
and
B
i
0
;
f
u
i
−
1
What
are
the
natures
of
the
triangles
O
A
i
B
i
?
c
Place
the
n
um
b
ers
u
4
and
u
5
on
the
x-axis
defined
b
y
the
relations
:
f
(
u
3
)
=
u
4
;
f
(
u
4
)
=
u
5
4
Generation
of
the
terms
of
the
sequence
:
a
En
ter
and
execute
this
program
in
the
programming
language
of
y
our
c
hoice.
x
←
0.25
For
i
ranging
from
0
to
100
x
←
5
4
−
1
x
+
1
End
For
A
t
the
end
of
execution,
what
is
the
v
alue
of
the
v
ari-
able
x
?
b
What
conjecture
can
b
e
made
ab
out
the
terms
of
this
sequence?
E.5107
Consider
the
sequence
u
n
n
∈
N
defined
b
y
:
u
0
=
5
;
u
n
=
1
+
2
n
·
u
n
−
1
+
6
n
p
our
n
∈
N
∗
.
1
a
Complete
the
follo
wing
table
:
n
0
1
2
3
4
5
6
u
n
b
Mak
e
a
conjecture
ab
out
the
nature
of
the
sequence
d
n
defined
b
y
:
d
n
=
u
n
+1
−
u
n
2
W
e
consider
the
sequence
v
n
defined
b
y
:
v
n
=
4
n
2
+
12
n
+
5
for
all
n
∈
N
∗
a
Giv
e
the
simplified
expression
of
the
expression
v
n
+1
as
a
function
of
n
.
b
Simplify
the
expression
of
:
1+
2
n
+1
·
v
n
+
6
n
+1
.
(W
e’l
l
use
the
factorization
:
4
x
3
+
24
x
2
+
41
x
+
21
=
(
x
+
1)(4
x
2
+
20
x
+
21)
)
c
What
can
w
e
sa
y
ab
out
the
sequences
u
n
et
v
n
.
h
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