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4
3
2
1
Outside the high school program
/ Sequels and variations
35 exercises (100% corrected)
a
Study of variations
(24 exercices)
Sequels and threshold search
(7 exercices)
Auxiliary sequences: variations
(3 exercices)
1.
Study
of
variations
E.7808
Consider
the
sequence
u
n
defined
for
an
y
in
teger
n
p
ositiv
e
or
zero
b
y
:
u
n
=
−
8
·
n
2
+
5
·
n
+
7
1
Establish
that
the
term
of
rank
n
+1
of
the
sequence
u
n
admits
as
expression
:
u
n
+1
=
−
8
·
n
2
−
11
·
n
+
4
2
By
studying
the
difference
u
n
+1
−
u
n
of
t
w
o
consecutiv
e
terms
of
the
sequence
u
n
,
sho
w
that
the
sequence
u
n
is
decreasing.
E.7809
Consider
the
sequence
u
n
defined
for
an
y
in
teger
n
p
ositiv
e
or
zero
b
y
:
u
n
=
−
7
·
n
2
+
6
·
n
+
1
1
Establish
that
the
term
of
rank
n
+1
of
the
sequence
u
n
admits
as
expression
:
u
n
+1
=
−
7
·
n
2
−
8
·
n
2
By
studying
the
difference
u
n
+1
−
u
n
of
t
w
o
consecutiv
e
terms
of
the
sequence
u
n
,
sho
w
that
the
sequence
u
n
is
decreasing.
E.7812
Consider
the
sequence
u
n
defined
for
an
y
strictly
p
ositiv
e
in
teger
n
b
y
:
u
n
=
3
·
n
−
1
n
−
1
2
By
studying
the
difference
of
t
w
o
consecutiv
e
terms,
sho
w
that
the
sequence
u
n
is
strictly
decreasing.
E.7824
Consider
the
sequence
u
n
defined
for
an
y
strictly
p
ositiv
e
in
teger
n
b
y
:
u
n
=
n
2
+1
n
+1
By
studying
the
difference
of
t
w
o
consecutiv
e
terms,
sho
w
that
the
sequence
u
n
is
strictly
decreasing.
E.7826
Consider
the
sequence
u
n
defined
for
an
y
in
teger
n
p
ositiv
e
or
zero
(
n
∈
N
)
b
y
the
relation
:
u
n
=
3
·
n
2
−
2
·
n
+
4
1
Express
the
term
u
n
+1
in
terms
of
n
.
2
By
studying
the
difference
u
n
+1
−
u
n
of
t
w
o
consecutiv
e
terms
of
the
sequence
u
n
,
sho
w
that
the
sequence
u
n
is
increasing.
E.7785
Consider
the
sequence
u
n
defined
for
all
natural
n
um
b
ers
n
b
y
the
relation
:
u
n
=
n
2
−
1
n
+
2
1
Determine
the
exact
v
alue
of
the
first
fiv
e
terms
of
the
sequence
u
n
.
Then,
complete
the
table
b
elo
w
with
the
v
alues
rounded
to
t
w
o
decimal
places
:
n
0
1
2
3
4
u
n
2
F
or
an
y
natural
n
um
b
er
n
,
sho
w
that
:
u
n
+1
−
u
n
=
n
2
+
5
·
n
+
3
n
+
2
n
+
3
3
Determine
the
direction
of
v
ariation
of
the
sequence
u
n
on
N
.
E.7778
Consider
the
sequence
u
n
defined
for
an
y
in
teger
n
p
ositiv
e
or
zero
(
n
∈
N
)
b
y
the
relation
:
u
n
=
400
×
0.8
n
+
30
1
Establish,
for
an
y
natural
n
um
b
er
n
(
n
∈
N
)
,
w
e
ha
v
e
the
relation
:
u
n
+1
−
u
n
=
−
80
×
0.8
n
2
Deduce
that
the
sequence
u
n
is
decreasing
on
N
.
E.7777
Consider
the
sequence
u
n
defined
for
an
y
in
teger
n
p
ositiv
e
or
zero
(
n
∈
N
)
b
y
the
relation
:
u
n
=
1.2
n
+
30
1
Establish,
for
an
y
natural
n
um
b
er
n
(
n
∈
N
)
,
w
e
ha
v
e
the
relation
:
u
n
+1
−
u
n
=
0.2
×
1.2
n
2
Deduce
that
the
sequence
u
n
is
increasing
on
N
.
E.7780
The
Asian
elephan
t
sp
ecies
are
endan-
gered.
The
p
opulation
in
2017
is
estimated
at
415
000
indi-
viduals.
The
n
um
b
er
of
elephan
ts
is
estimated
to
decrease
b
y
3
%
p
er
y
ear.
W
e
note
u
n
the
n
um
b
er
of
Asian
elephan
ts
in
the
y
ear
2017+
n
.
1
Giv
e
the
nature
and
c
haracteristic
elemen
ts
of
the
se-
quence
u
n
.
2
a
Giv
e
the
explicit
form
ula
for
the
rank
term
n
of
the
sequence
u
n
.
b
Giv
e
the
n
um
b
er
of
Asian
elephan
ts
in
2020
,
rounded
to
the
nearest
unit.
3
a
Giv
e
the
direction
of
v
ariation
of
the
sequence
u
n
.
Justify
y
our
answ
er.
b
Using
the
calculator,
determine
in
whic
h
y
ear
the
n
um-
b
er
of
Asian
elephan
ts
will
b
e
less
than
200
000
.
E.7761
Consider
the
sequence
u
n
defined
for
an
y
natural
n
um
b
er
n
(
n
∈
N
)
b
y
the
relation
:
u
n
=
1
2
·
n
+
1
1
Giv
e
the
v
alue
of
the
first
four
terms
of
the
sequence
u
n
.
2
a
Establish
the
iden
tit
y
for
an
y
natural
n
um
b
er
n
:
u
n
+1
−
u
n
=
−
2
2
·
n
+
1
2
·
n
+
3
b
Deduce
that
the
sequence
u
n
is
decreasing
on
N
.
E.7760
Consider
the
sequence
u
n
defined
for
all
natural
n
um
b
ers
n
(
n
∈
N
)
b
y
the
relation
:
u
n
=
n
+
3
n
+
1
1
Giv
e
the
v
alue
of
the
first
four
terms
of
the
sequence
u
n
.
2
a
Establish
the
iden
tit
y
for
an
y
natural
n
um
b
er
n
:
u
n
+1
−
u
n
=
−
2
n
+
1
n
+
2
b
Deduce
that
the
sequence
u
n
is
decreasing
on
N
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7808
sacados/7808
chapExoCorrec/7809
sacados/7809
chapExoCorrec/7812
sacados/7812
chapExoCorrec/7824
sacados/7824
chapExoCorrec/7826
sacados/7826
chapExoCorrec/7785
sacados/7785
chapExoCorrec/7778
sacados/7778
chapExoCorrec/7777
sacados/7777
chapExoCorrec/7780
sacados/7780
chapExoCorrec/7761
sacados/7761
chapExoCorrec/7760
sacados/7760
E.7756
Reminders
:
for
all
real
n
um
b
ers
a
and
b
and
for
all
relativ
e
in
tegers
n
and
m
,
w
e
ha
v
e
:
a
0
=
1
a
1
=
a
a
n
×
a
m
=
a
n
+
m
a
n
a
m
=
a
n
−
m
(
a
=0
)
a
n
m
=
a
n
×
m
a
n
×
b
n
=
a
×
b
n
a
n
b
n
=
a
b
n
(
b
=0
)
In
this
exercise,
w
e
will
highligh
t
the
monotonicit
y
of
the
se-
quences
using
the
quotien
t
metho
d.
1
The
sequence
u
n
n
∈
N
is
defined
b
y
:
u
n
=
1
2
n
for
all
n
∈
N
Sho
w
that
u
n
is
strictly
decreasing.
2
Consider
the
sequence
v
n
n
∈
N
defined
b
y
:
v
n
=
3
n
4
for
all
n
∈
N
.
Sho
w
that
v
n
is
strictly
increasing.
E.7762
Let
v
n
n
∈
N
b
e
the
sequence
defined
b
y
the
follo
wing
recurrence
relation
:
v
n
+1
=
v
n
−
v
n
2
−
1
for
all
n
∈
N
and
the
initial
condition
v
0
=
2
.
1
Using
a
calculator,
complete
the
table
b
elo
w
:
n
0
1
2
3
4
u
n
2
By
studying
the
difference
b
et
w
een
t
w
o
consecutiv
e
terms,
sho
w
that
the
sequence
v
n
is
decreasing.
E.7757
Study
the
sense
of
v
ariation
of
eac
h
of
the
follo
wing
sequences
:
1
The
sequence
u
n
is
defined
b
y
the
explicit
form
ula
:
u
n
=
3
2
n
for
all
n
∈
N
2
The
sequence
v
n
is
defined
b
y
the
explicit
form
ula
:
v
n
=
3
2
n
for
all
n
∈
N
E.4585
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
a
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
on
w
ards.
E.4586
In
this
exercise,
w
e
will
use
the
difference
metho
d
to
pro
v
e
the
monotonicit
y
of
the
sequences
:
1
Soit
u
n
n
∈
N
the
sequence
whose
rank
term
n
is
defined
b
y
:
u
n
=
−
32
n
+
102
Sho
w
that
this
sequence
is
decreasing.
2
Let
v
n
n
∈
N
∗
b
e
the
sequence
whose
term
of
rank
n
is
defined
b
y
:
v
n
=
2
n
−
1
Sho
w
that
this
sequence
is
increasing.
3
Let
w
n
n
∈
N
∗
b
e
the
sequence
whose
rank
term
n
is
de-
fined
b
y
:
w
n
=
2
n
−
25
n
Sho
w
that
the
sequence
w
n
is
increasing.
E.7758
Consider
the
arithmetic
sequence
u
n
of
first
term
5
and
reason
2
.
1
Giv
e
the
first
four
terms
of
the
sequence
u
n
.
2
Express
the
v
alue
of
the
term
u
n
as
a
function
of
its
rank
n
.
3
Sho
w
that
the
sequence
u
n
is
increasing.
E.7759
Consider
the
geometric
sequence
v
n
of
first
term
24
and
reason
1
2
.
1
Giv
e
the
first
four
terms
of
the
sequence
v
n
.
2
Express
the
v
alue
of
the
term
v
n
as
a
function
of
its
rank
n
.
3
Sho
w
that
the
sequence
v
n
is
decreasing.
E.7779
A
compan
y
decides
to
go
public.
When
it
go
es
public,
the
price
of
one
share
is
50
e
.
It
hop
es
that
its
share
price
will
increase
b
y
5
%
p
er
y
ear.
Let
u
0
b
e
the
share
price
when
it
go
es
public
and
u
n
,
for
an
y
strictly
p
ositiv
e
in
teger
n
,
the
share
price
after
n
y
ears.
1
Giv
e
the
nature
and
c
haracteristic
elemen
ts
of
the
se-
quence
u
n
.
2
a
Giv
e
the
explicit
form
ula
for
the
rank
term
n
of
the
sequence
u
n
.
b
Giv
e
the
share
price
after
10
y
ears
3
a
Giv
e
the
direction
of
v
ariation
of
the
sequence
u
n
.
Justify
y
our
answ
er.
b
Using
the
calculator,
determine
after
ho
w
man
y
y
ears
the
share
price
will
exceed
100
e
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7756
sacados/7756
chapExoCorrec/7762
sacados/7762
chapExoCorrec/7757
sacados/7757
chapExoCorrec/4585
sacados/4585
chapExoCorrec/4586
sacados/4586
chapExoCorrec/7758
sacados/7758
chapExoCorrec/7759
sacados/7759
chapExoCorrec/7779
sacados/7779
2
3
4
5
6
I
2
3
4
J
O
C
f
E.7763
1
Consider
the
sequence
u
n
defined
for
an
y
in
teger
n
p
os-
itiv
e
or
zero
(
n
∈
N
)
b
y
the
relation
:
u
n
=
3
·
n
+
5
a
Establish
that
the
term
of
rank
n
+1
admits
as
expres-
sion
:
u
n
+1
=
3
·
n
+
8
b
By
studying
the
sign
of
u
n
+1
−
u
n
,
sho
w
that
the
se-
quence
u
n
is
increasing.
2
Consider
the
sequence
v
n
defined
for
an
y
in
teger
n
p
os-
itiv
e
or
zero
(
n
∈
N
)
b
y
the
relation
:
v
n
=
−
2
·
n
2
+
n
+
2
a
Establish
that
the
term
of
rank
n
+1
admits
for
expres-
sion
:
v
n
+1
=
−
2
·
n
2
−
3
·
n
+
1
b
Establish
that
the
consecutiv
e
difference
of
t
w
o
terms
of
the
sequence
v
n
has
the
expression
:
v
n
+1
−
v
n
=
−
4
·
n
−
1
c
Deduce
that
the
sequence
v
n
is
a
decreasing
sequence
on
N
.
E.4619
Consider
the
function
f
defined
on
R
+
b
y
the
relation
:
f
(
x
)
=
6
x
−
2
2
x
+
1
In
the
co
ordinate
system
O
;
I
;
J
orthonormal
co
ordinate
system,
w
e
consider
the
curv
e
C
f
represen
ting
the
function
f
:
The
line
Δ
is
the
first
bisector
of
the
plane
;
its
Cartesian
equation
is
y
=
x
.
1
a
Establish
the
equalit
y
:
f
(
x
)
−
x
=
−
2
x
2
+
5
x
−
2
2
x
+
1
b
Determine
the
sign
table
for
the
expression
f
(
x
)
−
x
o
v
er
the
in
terv
al
0
;
+
∞
.
c
Deduce
the
relativ
e
p
ositions
of
the
curv
es
C
f
and
Δ
on
0
;
+
∞
.
2
Consider
the
sequence
u
n
définie
b
y
:
u
0
=
6
;
u
n
+1
=
f
u
n
a
Represen
t,
on
the
abscissa
axis
of
the
reference
frame,
the
first
six
terms
of
the
sequence
u
n
(the
c
onstruc-
tion
lines
wil
l
b
e
left
visible)
.
b
Determine
the
v
alue
of
the
first
three
terms
of
the
se-
quence
u
n
c
All
terms
in
the
sequence
are
assumed
to
b
e
greater
than
or
equal
to
2
.
Deduce
that
the
sequence
u
n
is
decreasing.
E.4607
F
or
eac
h
question,
determine,
b
y
study-
ing
the
difference
u
n
+1
−
u
n
,
the
direction
of
v
ariation
of
the
sequence
u
n
defined
b
y
:
a
u
n
=
3
n
2
+
n
+
1
b
u
n
=
2
n
+
3
n
−
1
c
u
n
+1
=
u
n
+
2
n
+
1
;
u
0
=
−
2
d
u
n
+1
=
u
n
−
n
+
5
;
u
0
=
2
E.4606
F
or
eac
h
question,
use
the
calculator
to
mak
e
a
conjecture
ab
out
the
direction
of
v
ariation
of
the
sequence
u
n
on
N
defined
b
y
:
a
u
n
=
n
3
−
2
n
2
−
3
n
b
u
n
=
5
n
n
+
2
c
u
n
=
3
1
+
(
−
1)
n
+
4
d
u
n
=
2
n
2
+
1
2
n
+
5
E.4605
F
or
eac
h
of
the
follo
wing
questions,
de-
termine
the
direction
of
v
ariation
of
the
sequence
u
n
n
∈
N
:
a
u
n
=
1
2
n
−
1
4
b
u
n
=
3
−
2
n
c
u
n
=
−
2
n
2
−
3
n
+
2
d
u
n
=
3
n
2
−
7
n
+
4
2.
Se
quels
and
thr
eshold
se
ar
ch
E.7204
A
shopk
eep
er
who
has
just
op
ened
a
b
ou-
tique
notices
that
his
sales
start
at
25
000
euros
p
er
mon
th
and
increase
ev
ery
mon
th
b
y
2
%
.
He
decides
to
mo
del
his
sales
progression
as
u
n
where
u
0
represen
ts
sales
when
the
store
op
ens.
1
Giv
e
the
nature
and
c
haracteristics
of
the
sequence
u
n
.
2
a
Using
the
calculator,
determine
after
ho
w
man
y
mon
ths
his
sales
will
exceed
30
000
euros.
b
Complete
the
algorithm
so
that,
at
the
end
of
its
exe-
cution,
the
v
ariable
n
has
the
v
alue
of
the
n
um
b
er
of
mon
ths
to
w
ait
b
efore
its
sales
exceed
30
000
euros.
‘
.1
n
←
0
‘
.
2
u
←
25
000
‘
.3
As
long
as
.
..
make
‘
.4
n
←
...
‘
.5
u
←
...
‘
.6
End
As
long
as
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7763
sacados/7763
chapExoCorrec/4619
sacados/4619
2
3
4
5
6
I
2
3
4
J
O
C
f
chapExoCorrec/4607
sacados/4607
chapExoCorrec/4606
sacados/4606
chapExoCorrec/4605
sacados/4605
chapExoCorrec/7204
sacados/7204
E.7209
In
order
to
com
bat
air
p
ollution,
as
early
as
the
y
ear
2013
certain
companies
w
ere
obliged
to
reduce
the
quan
tit
y
of
p
olluting
pro
ducts
they
released
in
to
the
air
eac
h
y
ear.
These
companies
disc
harged
410
tonnes
of
these
p
ollutan
ts
in
2013
and
332
tonnes
in
2015
.
The
ann
ual
rate
of
decrease
in
the
mass
of
p
ollutan
ts
released
is
assumed
to
b
e
constan
t.
1
Justify
that
the
y
ear-on-y
ear
c
hange
can
b
e
considered
to
corresp
ond
to
a
decrease
of
10
%
.
2
This
rate
of
10
%
is
assumed
to
remain
constan
t
for
the
coming
y
ears.
a
Using
the
calculator,
determine
from
whic
h
y
ear
on-
w
ards
the
quan
tit
y
of
p
ollutan
ts
disc
harged
b
y
these
companies
will
no
longer
exceed
the
threshold
of
180
tonnes
set
b
y
the
departmen
tal
council.
b
Complete
the
algorithm
b
elo
w
so
that
the
v
ariable
n
has,
at
the
end
of
its
execution,
the
v
alue
of
the
y
ear
in
whic
h
the
quan
tit
y
of
p
ollutan
ts
disc
harged
will
not
exceed
180
tonnes.
n
←
0
u
←
410
As
long
as
...
n
←
n+1
u
←
u
×
0.9
End
As
long
as
n
←
n+...
E.7600
Consider
the
geometric
sequence
u
n
with
first
term
1
and
common
ratio
2
.
Consider
the
co
de
:
Function
f(n)
u
←
1
For
i
ranging
from
1
to
n
u
←
2
×
u
End
For
Return
u
The
function
call
f(n)
returns
the
v
alue
of
the
term
of
the
sequence
u
n
of
rank
n
to
the
program.
Complete
the
table
of
v
alues
b
elo
w
:
n
0
1
2
10
20
u
n
E.7569
A
w
ebsite
offers
its
subscrib
ers
mo
vies
to
do
wnload.
When
it
op
ens,
500
films
are
offered
and
eac
h
mon
th
the
n
um-
b
er
of
films
offered
to
subscrib
ers
increases
b
y
6
%
.
W
e
mo
del
the
n
um
b
er
of
films
offered
b
y
a
sequence
u
n
where
n
denotes
the
n
um
b
er
of
mon
ths
since
the
site
op
ened.
1
Calculate
u
0
,
u
1
and
u
2
and
giv
e
the
result
rounded
to
the
nearest
unit.
2
Giv
e
the
nature
and
c
haracteristic
elemen
ts
of
the
se-
quence
u
n
.
Express
u
n
as
a
function
of
n
.
3
Determine
the
v
alue
of
the
rank
term
6
rounded
to
unit
y
.
4
Using
the
calculator,
determine
after
ho
w
man
y
mon
ths
the
n
um
b
er
of
films
offered
exceeds
800
films
offered.
E.7570
A
small
to
wn
has
a
m
unicipal
bicycle
ren
tal
service.
The
m
unicipalit
y
w
ould
lik
e
information
on
the
n
um
b
er
of
bicycles
in
circulation
and
the
cost
in
v
olv
ed.
The
manager
of
the
bik
e
ren
tal
service
notes
that
b
et
w
een
un-
usable
bik
es,
b
ecause
lost,
stolen
or
damaged,
and
new
bik
es
acquired,
the
n
um
b
er
of
usable
bik
es
increases
b
y
5
%
ev
ery
y
ear.
On
1
er
Jan
uary
2017
,
the
fleet
con
tains
200
usable
bik
es.
W
e
mo
del
the
ev
olution
of
the
n
um
b
er
of
usable
bik
es
b
y
a
sequence
u
n
in
whic
h,
for
an
y
natural
in
teger
n
,
u
n
is
the
n
um
b
er
of
bicycles
on
1
er
Jan
uary
of
the
y
ear
2017+
n
.
Th
us,
u
0
=
200
and,
for
an
y
natural
n
um
b
er
n
:
u
n
+1
=
1.05
×
u
n
.
1
a
Justify
the
co
efficien
t
1.05
in
the
expression
of
u
n
+1
as
a
function
of
u
n
.
b
Ho
w
man
y
bicycles
will
there
b
e
in
this
park
at
1
er
Jan
uary
2018
?
2
The
m
unicipalit
y
has
decided
to
stop
buying
new
bik
es
as
so
on
as
its
sto
c
k
exceeds
500
units.
In
whic
h
y
ear
will
the
m
unicipal
service’s
sto
c
k
exceed
500
bik
es
for
the
first
time?
E.7599
Consider
the
geometric
sequence
u
n
,
with
common
ratio
0.9
and
first
term
u
0
=
50
.
1
a
Cop
y
and
complete
the
algorithm
so
that,
at
the
end
of
its
execution,
the
v
ariable
U
has
the
v
alue
25
e
,
whic
h
is
the
term
of
this
sequence,
i.e.,
u
24
:
U
←
...
For
N
ranging
from
1
to
24
U
←
...
End
For
b
F
or
an
y
natural
n
um
b
er
n
,
express
u
n
in
terms
of
n
.
c
Calculate
u
24
,
then
giv
e
its
v
alue
rounded
to
the
near-
est
10
−
3
.
2
Determine
the
smallest
natural
n
um
b
er
n
suc
h
that
:
u
n
<
0.01
.
E.7595
A
store
offers
a
lo
y
alt
y
card
to
enjo
y
b
enefits
when
making
purc
hases.
In
the
first
y
ear,
the
lo
y
alt
y
card
w
as
offered
to
200
customers.
W
e
observ
e
that
the
n
um
b
er
of
customers
with
the
lo
y
alt
y
card
increases
b
y
6
%
p
er
y
ear.
W
e
mo
del
the
n
um
b
er
of
lo
y
alt
y
card
holders
b
y
a
sequence
u
n
where
n
denotes
the
n
um
b
er
of
y
ears
since
the
store
op
ened.
1
Calculate
u
0
,
u
1
and
u
2
and
giv
e
the
result
rounded
to
unit
y
.
2
Giv
e
the
nature
of
the
sequence
and
its
c
haracteristic
elemen
ts.
Express
u
n
as
a
function
of
n
.
3
Determine
the
v
alue
of
the
rank
term
6
,
rounded
to
the
nearest
unit.
4
Using
the
calculator,
determine
after
ho
w
man
y
y
ears
the
n
um
b
er
of
lo
y
alt
y
card
holders
exceeds
400
p
eople.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7209
sacados/7209
chapExoCorrec/7600
sacados/7600
chapExoCorrec/7569
sacados/7569
chapExoCorrec/7570
sacados/7570
chapExoCorrec/7599
sacados/7599
chapExoCorrec/7595
sacados/7595
3.
A
uxiliary
se
quenc
es
:
variations
E.6017
Consider
the
sequence
u
n
defined
b
y
:
u
0
=
7
;
u
n
+1
=
1
3
·
u
n
+
4
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
=
u
n
−
6
a
Establish
the
equalit
y
b
elo
w
for
an
y
natural
n
um
b
er
n
:
v
n
+1
=
1
3
·
v
n
b
Giv
e
the
first
term
of
the
sequence
v
n
.
c
Giv
e
the
direction
of
v
ariation
of
the
sequence
v
n
.
2
Deduce
the
direction
of
v
ariation
of
the
sequence
u
n
.
E.6042
Consider
the
sequence
u
n
defined
on
N
b
y
the
relation
:
u
0
=
1
;
u
n
+1
=
1
2
·
u
n
−
3
for
all
n
∈
N
1
Define
the
sequence
v
n
on
N
b
y
the
relation
:
v
n
=
2
·
u
n
+
12
a
Pro
v
e
the
relation
:
v
n
+1
=
1
2
·
v
n
for
all
n
∈
N
.
b
Giv
e
the
expression
of
the
terms
of
the
sequence
v
n
in
terms
of
n
.
Justify
y
our
approac
h.
2
a
Justify
that
for
an
y
natural
n
um
b
er
n
,
w
e
ha
v
e
:
u
n
=
14
·
1
2
n
+1
−
6
b
Deduce
the
direction
of
v
ariation
of
the
sequence
u
n
E.6063
Let
u
n
b
e
the
n
umerical
sequence
defined
on
N
b
y
:
u
0
=
3
;
u
n
+1
=
2
·
u
n
−
n
+
1
for
n
∈
N
.
1
Calculate
u
1
,
u
2
and
u
3
.
2
W
e
denote
b
y
v
n
the
sequence
defined
on
N
b
y
:
v
n
=
u
n
−
n
a
Justify
that
for
an
y
natural
n
um
b
er
n
,
w
e
ha
v
e
the
re-
lationship
:
v
n
+1
=
2
·
v
n
.
b
Determine
the
nature
and
c
haracteristic
elemen
ts
of
the
sequence
v
n
.
3
a
Justify
that
for
an
y
natural
n
um
b
er
n
,
w
e
ha
v
e
:
u
n
=
3
×
2
n
+
n
.
b
Deduce
the
direction
of
v
ariation
of
the
sequence
u
n
.
4.
Unclassifie
d
financial
ye
ars
E.5817
Consider
the
sequence
u
n
de-
fined
b
y
its
first
term
u
1
=
3
2
and
the
recurrence
relation
:
u
n
+1
=
n
·
u
n
+
1
2(
n
+
1)
p
our
tout
n
∈
N
∗
W
e
define
an
auxiliary
sequence
v
n
b
y
:
v
n
=
n
·
u
n
−
1
for
an
y
in
teger
n
1
1
Sho
w
that
the
sequence
v
n
is
geometric
;
sp
ecify
its
reason
and
first
term.
2
Deduce
that,
for
an
y
natural
n
um
b
er
n
1
,
w
e
ha
v
e
:
u
n
=
1
+
(0.5)
n
n
3
Determine
the
limit
of
the
sequence
u
n
.
4
Justify
that,
for
an
y
in
teger
n
1
,
w
e
ha
v
e
:
u
n
+1
−
u
n
=
−
1
+
(1
+
0.5
n
)
·
(0
;
5)
n
n
·
(
n
+
1)
Deduce
the
direction
of
v
ariation
of
the
sequence
u
n
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6017
sacados/6017
chapExoCorrec/6042
sacados/6042
chapExoCorrec/6063
sacados/6063
chapExoCorrec/5817
sacados/5817
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