التمرين الأول: ( 5ن )
لتكن
f
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadAgaaaa@36EF@
الدالة العددية المعرفة بما
يلي:
f(
x
)=
x
2
−2xsinx+1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadAgadaqada
qaaiaadIhaaiaawIcacaGLPaaacqGH9aqpdaGcaaqaaiaadIhadaah
aaWcbeqaaiaaikdaaaGccqGHsislcaaIYaGaamiEaiGacohacaGGPb
GaaiOBaiaadIhacqGHRaWkcaaIXaaaleqaaaaa@449E@
.
(1ن) 1)- تحقق أن
D
f
=ℝ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadseadaWgaa
WcbaGaamOzaaqabaGccqGH9aqpcqWIDesOaaa@3A66@
و أن
f
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadAgaaaa@36EF@
متصلة على
ℝ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiabl2riHcaa@3776@
.
(1ن) 2)- أحسب النهايتين
lim
x→+∞
f(
x
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaaxababaGaci
iBaiaacMgacaGGTbaaleaacaWG4bGaeyOKH4Qaey4kaSIaeyOhIuka
beaakiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaaaaa@41C7@
و
lim
x→+∞
f(
x
)
x
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaaxababaGaci
iBaiaacMgacaGGTbaaleaacaWG4bGaeyOKH4Qaey4kaSIaeyOhIuka
beaakmaalaaabaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaa
qaaiaadIhaaaaaaa@42D4@
.
(1.5ن) 3)- أحسب
lim
x→1
sinπx
x−1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaaxababaGaci
iBaiaacMgacaGGTbaaleaacaWG4bGaeyOKH4QaaGymaaqabaGcdaWc
aaqaaiGacohacaGGPbGaaiOBaiabec8aWjaadIhaaeaacaWG4bGaey
OeI0IaaGymaaaaaaa@4505@
و استنتج
lim
x→0
sin(
πf(
x
)
)
x
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaaxababaGaci
iBaiaacMgacaGGTbaaleaacaWG4bGaeyOKH4QaaGimaaqabaGcdaWc
aaqaaiGacohacaGGPbGaaiOBamaabmaabaGaeqiWdaNaamOzamaabm
aabaGaamiEaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaqaaiaadIha
aaaaaa@4759@
.
(1.5ن) 4)- بين أن:
∀n∈
ℕ
∗
,
lim
x→+∞
f
[ n ]
(
x
)
x
=1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiabgcGiIiaad6
gacqGHiiIZcqWIvesPdaahaaWcbeqaaiabgEHiQaaakiaacYcadaWf
qaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamiEaiabgkziUkabgUcaRi
abg6HiLcqabaGcdaWcaaqaaiaadAgadaahaaWcbeqaamaadmaabaGa
amOBaaGaay5waiaaw2faaaaakmaabmaabaGaamiEaaGaayjkaiaawM
caaaqaaiaadIhaaaGaeyypa0JaaGymaaaa@4E3A@
،حيث
f
[ n ]
=f∘f∘...∘f
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgadaahaa
WcbeqaamaadmaabaGaamOBaaGaay5waiaaw2faaaaakiabg2da9iaa
dAgacqWIyiYBcaWGMbGaeSigI8MaaiOlaiaac6cacaGGUaGaeSigI8
MaamOzaaaa@4398@
..
التمرين الثاني: ( 7ن )
لتكن
f
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadAgadaWgaa
WcbaGaamOBaaqabaaaaa@380E@
الدالة المعرفة على
ℝ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiabl2riHcaa@3774@
بما يلي:
f
n
(
x
)=
x
3
+(
n+1
)x−1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadAgadaWgaa
WcbaGaamOBaaqabaGcdaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH
9aqpcaWG4bWaaWbaaSqabeaacaaIZaaaaOGaey4kaSYaaeWaaeaaca
WGUbGaey4kaSIaaGymaaGaayjkaiaawMcaaiaadIhacqGHsislcaaI
Xaaaaa@4535@
،
(
n∈ℕ
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaamaabmaabaGaam
OBaiabgIGiolablwriLcGaayjkaiaawMcaaaaa@3B70@
(1ن) 1)- بين أن المعادلة
f
n
(
x
)=0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadAgadaWgaa
WcbaGaamOBaaqabaGcdaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH
9aqpcaaIWaaaaa@3C5E@
تقبل حلا وحيدا
α
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiabeg7aHnaaBa
aaleaacaWGUbaabeaaaaa@38C2@
في
ℝ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiabl2riHcaa@3776@
و أن
0<
α
n
<1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaaicdacqGH8a
apcqaHXoqydaWgaaWcbaGaamOBaaqabaGccqGH8aapcaaIXaaaaa@3C49@
.
(1ن) 2)- بين أن المتتالية
(
α
n
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaamaabmaabaGaeq
ySde2aaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaaaaa@3A55@
تناقصية ،و استنتج أنها متقاربة.
(1ن) 3)- بين أن
∀n∈ℕ,0<
α
n
<
1
n+1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiabgcGiIiaad6
gacqGHiiIZcqWIvesPcaGGSaGaaGimaiabgYda8iabeg7aHnaaBaaa
leaacaWGUbaabeaakiabgYda8maalaaabaGaaGymaaqaaiaad6gacq
GHRaWkcaaIXaaaaaaa@444C@
، ثم استنتج
lim
n→+∞
α
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaaxababaGaci
iBaiaacMgacaGGTbaaleaacaWGUbGaeyOKH4Qaey4kaSIaeyOhIuka
beaakiabeg7aHnaaBaaaleaacaWGUbaabeaaaaa@410A@
.
(1ن) 4)- أحسب النهايتين:
lim
n→+∞
n
α
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaaxababaGaci
iBaiaacMgacaGGTbaaleaacaWGUbGaeyOKH4Qaey4kaSIaeyOhIuka
beaakiaad6gacqaHXoqydaWgaaWcbaGaamOBaaqabaaaaa@41FD@
و
lim
n→+∞
n
α
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaaxababaGaci
iBaiaacMgacaGGTbaaleaacaWGUbGaeyOKH4Qaey4kaSIaeyOhIuka
beaakmaakaaabaGaamOBaaWcbeaakiabeg7aHnaaBaaaleaacaWGUb
aabeaaaaa@4222@
.
(1ن) 5)-أ- بين أن:
0≤
α
p
−
α
q
≤q
α
q
−p
α
p
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaaicdacqGHKj
YOcqaHXoqydaWgaaWcbaGaamiCaaqabaGccqGHsislcqaHXoqydaWg
aaWcbaGaamyCaaqabaGccqGHKjYOcaWGXbGaeqySde2aaSbaaSqaai
aadghaaeqaaOGaeyOeI0IaamiCaiabeg7aHnaaBaaaleaacaWGWbaa
beaaaaa@490F@
، لكل
p
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadchaaaa@36FB@
و
q
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadghaaaa@36FC@
من
ℕ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiablwriLcaa@3772@
حيث
p≤q
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadchacqGHKj
YOcaWGXbaaaa@39A6@
.
(1ن) ب- بين أن:
0≤
α
p
−
α
q
≤
q−p
p+1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaaicdacqGHKj
YOcqaHXoqydaWgaaWcbaGaamiCaaqabaGccqGHsislcqaHXoqydaWg
aaWcbaGaamyCaaqabaGccqGHKjYOdaWcaaqaaiaadghacqGHsislca
WGWbaabaGaamiCaiabgUcaRiaaigdaaaaaaa@4626@
، لكل
p
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadchaaaa@36FB@
و
q
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadghaaaa@36FC@
من
ℕ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiablwriLcaa@3772@
حيث
p≤q
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadchacqGHKj
YOcaWGXbaaaa@39A6@
.
(1ن) ج- استنتج أن:
∀n∈ℕ,0≤
α
n
−
α
n+1
≤
1
n+1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiabgcGiIiaad6
gacqGHiiIZcqWIvesPcaGGSaGaaGimaiabgsMiJkabeg7aHnaaBaaa
leaacaWGUbaabeaakiabgkHiTiabeg7aHnaaBaaaleaacaWGUbGaey
4kaSIaaGymaaqabaGccqGHKjYOdaWcaaqaaiaaigdaaeaacaWGUbGa
ey4kaSIaaGymaaaaaaa@4B00@
.
التمرين الثالث: ( 8ن )
لتكن
(
a
n
)
n≥1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
yyamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaaBaaaleaa
caWGUbGaeyyzImRaaGymaaqabaaaaa@3D3E@
متتالية عددية و
(
b
n
)
n≥1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
OyamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaaBaaaleaa
caWGUbGaeyyzImRaaGymaaqabaaaaa@3D3F@
،
(
c
n
)
n≥1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
4yamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaaBaaaleaa
caWGUbGaeyyzImRaaGymaaqabaaaaa@3D40@
و
(
d
n
)
n≥1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
izamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaaBaaaleaa
caWGUbGaeyyzImRaaGymaaqabaaaaa@3D41@
متتاليات عددية بحيث:
b
n
=
a
n
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadkgadaWgaa
WcbaGaamOBaaqabaGccqGH9aqpdaWcaaqaaiaadggadaWgaaWcbaGa
amOBaaqabaaakeaacaWGUbaaaaaa@3C2E@
و
c
n
=
a
n+1
−
a
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadogadaWgaa
WcbaGaamOBaaqabaGccqGH9aqpcaWGHbWaaSbaaSqaaiaad6gacqGH
RaWkcaaIXaaabeaakiabgkHiTiaadggadaWgaaWcbaGaamOBaaqaba
aaaa@3FBB@
و
d
n
=
a
1
+
a
2
+...+
a
n
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadsgadaWgaa
WcbaGaamOBaaqabaGccqGH9aqpdaWcaaqaaiaadggadaWgaaWcbaGa
aGymaaqabaGccqGHRaWkcaWGHbWaaSbaaSqaaiaaikdaaeqaaOGaey
4kaSIaaiOlaiaac6cacaGGUaGaey4kaSIaamyyamaaBaaaleaacaWG
UbaabeaaaOqaaiaad6gaaaaaaa@449B@
لكل
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaad6gaaaa@36F9@
من
ℕ
∗
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiablwriLoaaCa
aaleqabaGaey4fIOcaaaaa@388E@
.
نرمز ب
α
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiabeg7aHbaa@37A5@
للحل الموجب للمعادلة
x
2
−x−1=0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadIhadaahaa
WcbeqaaiaaikdaaaGccqGHsislcaWG4bGaeyOeI0IaaGymaiabg2da
9iaaicdaaaa@3D48@
. (
α
2
=α+1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiabeg7aHnaaCa
aaleqabaGaaGOmaaaakiabg2da9iabeg7aHjabgUcaRiaaigdaaaa@3CDA@
).
(1ن) 1)-أ- بين أنه
إذا كانت
(
a
n
)
n≥1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
yyamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaaBaaaleaa
caWGUbGaeyyzImRaaGymaaqabaaaaa@3D3E@
متقاربة و
lim
n→+∞
a
n
=l
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbmqadeWacmGadiqabmqadmWabmabdaGcbaWaaCbeaeaaci
GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHRaWkcqGHEisP
aeqaaOGaamyyamaaBaaaleaacaWGUbaabeaakiabg2da9iaadYgaaa
a@425D@
فإن
lim
n→+∞
d
n
=l
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaaxababaGaci
iBaiaacMgacaGGTbaaleaacaWGUbGaeyOKH4Qaey4kaSIaeyOhIuka
beaakiaadsgadaWgaaWcbaGaamOBaaqabaGccqGH9aqpcaWGSbaaaa@4255@
.
(1ن) ب- استنتج أنه إذا كانت
(
c
n
)
n≥1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
4yamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaaBaaaleaa
caWGUbGaeyyzImRaaGymaaqabaaaaa@3D40@
متقاربة و
lim
n→+∞
c
n
=l
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaaxababaGaci
iBaiaacMgacaGGTbaaleaacaWGUbGaeyOKH4Qaey4kaSIaeyOhIuka
beaakiaadogadaWgaaWcbaGaamOBaaqabaGccqGH9aqpcaWGSbaaaa@4254@
فإن
lim
n→+∞
b
n
=l
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaaxababaGaci
iBaiaacMgacaGGTbaaleaacaWGUbGaeyOKH4Qaey4kaSIaeyOhIuka
beaakiaadkgadaWgaaWcbaGaamOBaaqabaGccqGH9aqpcaWGSbaaaa@4253@
.
·
نفترض فيما
يلي أن:
a
1
=2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadggadaWgaa
WcbaGaaGymaaqabaGccqGH9aqpcaaIYaaaaa@399F@
و
a
n+1
=
a
n
+
1+
b
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadggadaWgaa
WcbaGaamOBaiabgUcaRiaaigdaaeqaaOGaeyypa0JaamyyamaaBaaa
leaacaWGUbaabeaakiabgUcaRmaakaaabaGaaGymaiabgUcaRiaadk
gadaWgaaWcbaGaamOBaaqabaaabeaaaaa@415C@
، لكل
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaad6gaaaa@36F9@
من
ℕ
∗
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiablwriLoaaCa
aaleqabaGaey4fIOcaaaaa@388E@
.
(1ن) 2)-أ- بين أن
المتتالية
(
a
n
)
n≥1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
yyamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaaBaaaleaa
caWGUbGaeyyzImRaaGymaaqabaaaaa@3D3E@
تزايدية.
(1ن) ب- بين بالترجع أن :
∀n∈
ℕ
∗
,
a
n
>nα
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiabgcGiIiaad6
gacqGHiiIZcqWIvesPdaahaaWcbeqaaiabgEHiQaaakiaacYcacaWG
HbWaaSbaaSqaaiaad6gaaeqaaOGaeyOpa4JaamOBaiabeg7aHbaa@4237@
، و استنتج
lim
n→+∞
a
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaaxababaGaci
iBaiaacMgacaGGTbaaleaacaWGUbGaeyOKH4Qaey4kaSIaeyOhIuka
beaakiaadggadaWgaaWcbaGaamOBaaqabaaaaa@4051@
.
(1ن) ج- بين أن
∀n∈
ℕ
∗
,
a
n
2
−n
a
n
−
n
2
>0
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiabgcGiIiaad6
gacqGHiiIZcqWIvesPdaahaaWcbeqaaiabgEHiQaaakiaacYcacaWG
HbWaa0baaSqaaiaad6gaaeaacaaIYaaaaOGaeyOeI0IaamOBaiaadg
gadaWgaaWcbaGaamOBaaqabaGccqGHsislcaWGUbWaaWbaaSqabeaa
caaIYaaaaOGaeyOpa4JaaGimaaaa@47DE@
.( يمكن استعمال إشارة
x
2
−x−1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadIhadaahaa
WcbeqaaiaaikdaaaGccqGHsislcaWG4bGaeyOeI0IaaGymaaaa@3B88@
).
(1ن) 3)- بين أن
المتتالية
(
b
n
)
n≥1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
OyamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaaBaaaleaa
caWGUbGaeyyzImRaaGymaaqabaaaaa@3D3F@
تناقصية، و أن
∀n∈
ℕ
∗
,
b
n
>α
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiabgcGiIiaad6
gacqGHiiIZcqWIvesPdaahaaWcbeqaaiabgEHiQaaakiaacYcacaWG
IbWaaSbaaSqaaiaad6gaaeqaaOGaeyOpa4JaeqySdegaaa@4146@
.
(1ن) 4)- بين أن
المتتالية
(
c
n
)
n≥1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
4yamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaaBaaaleaa
caWGUbGaeyyzImRaaGymaaqabaaaaa@3D40@
تناقصية، و أن
∀n∈
ℕ
∗
,
c
n
>α
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiabgcGiIiaad6
gacqGHiiIZcqWIvesPdaahaaWcbeqaaiabgEHiQaaakiaacYcacaWG
JbWaaSbaaSqaaiaad6gaaeqaaOGaeyOpa4JaeqySdegaaa@4147@
.
(1ن) 5)- استنتج أن
المتتاليتين
(
b
n
)
n≥1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
OyamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaaBaaaleaa
caWGUbGaeyyzImRaaGymaaqabaaaaa@3D3F@
و
(
c
n
)
n≥1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
4yamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaaBaaaleaa
caWGUbGaeyyzImRaaGymaaqabaaaaa@3D40@
متقاربتان، و حدد نهايتهما.