Bài toán lãi suất và tăng trưởng | Toán Cao Cấp | Trường Đại học Thủy Lợi

Bài toán lãi suất và tăng trưởng của Trường Đại học Thủy Lợi. Hi vọng tài liệu này sẽ giúp các bạn học tốt, ôn tập hiệu quả, đạt kết quả cao trong các bài thi, bài kiểm tra sắp tới. Mời các bạn cùng tham khảo chi tiết bài viết dưới đây nhé.

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Toán Cao Cấp (BLLLO2001) 14 tài liệu

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Đại học Thủy Lợi 221 tài liệu

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Bài toán lãi suất và tăng trưởng | Toán Cao Cấp | Trường Đại học Thủy Lợi

Bài toán lãi suất và tăng trưởng của Trường Đại học Thủy Lợi. Hi vọng tài liệu này sẽ giúp các bạn học tốt, ôn tập hiệu quả, đạt kết quả cao trong các bài thi, bài kiểm tra sắp tới. Mời các bạn cùng tham khảo chi tiết bài viết dưới đây nhé.

115 58 lượt tải Tải xuống

Môn: Toán Cao Cấp (BLLLO2001) 14 tài liệu

Trường: Đại học Thủy Lợi 221 tài liệu

Thông tin:

25 trang 4 tháng trước

Tác giả:

Profile Lê Phương
CHỦ ĐỀ 9: BÀI TOÁN LÃI SUẤT, TĂNG TRƯỞNG
II. CÁC DẠNG TOÁN TRỌNG TÂM VÀ PHƯƠNG PHÁP GIẢI
BÀI TOÁN 1. CÔNG THỨC LÃI KÉP
T = A(+r)
n
Ar
n
T !"#
$"%&"#L = − =T AA(+r)
n
− A
Ví dụ 1: [Đề thi THPT Quốc gia 2017] '(")*+,-./01(23
456178-98-:-;2<1=61>"#%?0
@A0618?B0C;A0D61")E%"#F,,-.
/0/1 GH I-0)")E-:-
A.J61 B.K61 C.L61 D.61
Lời giải
HMn∈
+
61@EF,,-./
N&-+,( +45)
n
>,,⇔ >nO⇒ =nK61Chọn B.
Ví dụ 2: [Đề thi THPT Quốc gia 2018] '(")*8.101(23P+561
78-98-:-;2<1=61>"#%?0@A
0618?B0C;A0D61")E"#Q *R?
* I-00 )&&")E-:-G
A.61 B.O61 C.,61 D.K61
Lời giải
S?TUV?ET = A(+r)
n
-0Er = P+5T ≥ KA
N&-A(+P+5)
n
≥ KA⇒,P+
n
≥ ⇔ ≥Kn 0
,P+
K≈ O+W
X%&A,61@")E"#?Chọn C.
Ví dụ 3: [Đề thi THPT Quốc gia 2017] Y61K,4Z%?1(&[Z
T\@- "F02D-061K,4]/78-91=61<T\
@- "F02D-0 61E6D1+50361-"3C;610T"32&61
D1ZT\@- "F02D-0 613FK]/G
A.$61K,KK B.$61K,K C.$61K,K, D.$61K,KJ
Lời giải
[Z- "F02Dn61T =T
,
(+r)
n
=+( + 5)
n
lOMoARcPSD|40651217
H (++ 5)
n
≥ ⇒ ≥K n LO+⇒ =n
+Chọn B.
Ví dụ 4: [Đề thi ở GD&ĐT Hà Nội năm 2017] ^ X.T_I*021(3
4+51(6178-91=61>"#%?0[A@x
Q-./x∈RX.*02@J61`11(8aBb1c&-IcJ,
-./
A.+,-./ B.+L-./ C.L+-./ D.L,-./
Lời giải
V?T = A(+r)
n
[X.EJ61>!-d
[EA A A -.Chọn C.
Ví dụ 5:N1()1.IZEL+,-./IZ?
*e2Z-N01B0?"FV?Ne?IZ
*e2Z-3K51(fg-0)WcNe?IZ
*e2N013,PJ51(c-0),c["#
e2+,,,+OK,J-./C;IZ*e1=2Z-
N010DG
A.KW,-.P,-. B.P,-.KW,-.
C.K,,-.K+,-. D.K+,-.K,,-.
Lời giải
HMx y Q-./R"#1IZ*02Z-N01
N1IZ%"#*02Z-t

= x( + K5)
4
− x-.
N1IZ%"#*02N01t
K
= y( +,PJ5)
,
− y-.
hEE.?"F-<x+ =( y+ L+,)4+ ( + ),= ⇒xy ==P,KW,Chọn A.
x K5 y ,PJ5 +,,,,+OK
Ví dụ 6: [Trích ề tham khảo của bộ GD&ĐT năm 2018] '( ")*,,-./01(2
3,L5c78-98-:-;2<1=c>
"#%?0@A0c8?B0C;:4c")E"#iQ
R30T"32&8-00 )&")E-:
-&G
A.,KLKL,,,/ B.,KLKJ,,,/
C.,K,4,,,/
D.,K,P,,,/
Lời giải
N")E%"#4c,,,,,,,,( +,L5)
4
=,KLKL,,,Chọn A.
BÀI TOÁN 2. CÔNG THỨC TĂNG TRƯỞNG DÂN SỐ
N = N
,
(+r)
n
-0EN
,
T261r].6T261n61N
T261<1
Ví dụ 1:[B0c0c0`A?`T2`"3A8cK61K,WO+OJ-.")
8].6-"eT2-<961JJ5<T2"30cK61K,K+0
DG
Lời giải
j2"30cK61K,K+N = O+OJJJ5( +)
P
≈,+KJ-.")
Ví dụ 2:j2`1(a.&,,,,")")T_0cK61kT2aE,L,L
")C;-<1=61T2`aE60D?-61
Lời giải
[B0EN = N
,
(+r)
n
⇒,L,L=,,,,( +r)
K
⇒ =r,,K561
BÀI TOÁN 3. HAO MÒN TÀI SẢN, DIỆN TÍCH RỪNG BỊ GIẢM…
WX Công thức hao mòn tài sản: H = H
,
(−r)
n
-0EH
,
c-I :Hc-I 
n61r].01lAB061
mnCông thức diện tích rừng bị giảm: T =T
,
(−r)
n
-0ET
,
T.A-dTT.A
-dn61r].-d 1961
Ví dụ 1:H *1(61T.A-d`"3 1x?-61T.A.EC;L
61T.A-d`"3>0D?T.A.&G
A.−,, x L B.,,5 C.−,, Lx D.−,, x L
Lời giải
N61n T.A-dlT
,
(−
r
)
n
DL61T.A-d>

−
,,
x

L
?
lOMoARcPSD|40651217
T.A"3.&Chọn D.
Ví dụ 2:'(")11(8aBNC-IcOW-./Ac-I`8aBE+618
-91=61c-I`8aB 1,5
Lời giải
Hc-I`8aB+61T = OW,( − 5)
+
= +PWP-./
Ví dụ 3:h1(10"#1E84,,°CV0`E 1+,5N10"#f
"o4,,°C8.(106D1+°C<(V0`E 1J+5.E7810
&E(V0KW,MPaT"34,,°C"#*TU-0.a2&T_cl.?$81
0@(V0`%.&JWMPa<.(0`l.?90
DAB0(BG
A.4K, B.4+ C.4,+ D.4,
Lời giải
Y(V0KW,MPaT"34,,°C Y84,,°CV0`E 1+,5lL,MPa 
$.(106+°C<(V0`E 1J+5DE
L, − ,,
J+

n
≥ JW⇔ ≤n J,KP
N&-n = J'=6+°C⇒J6+°C$.
(04+°C Chọn B.
BÀI TOÁN 4. TĂNG TRƯỞNG CỦA BÈO, CỦA VI KHUẨN…
WX Tăng trưởng của bèo:
H *"#p0T
,
1=)"#p06?K<n)"#p0>
T =T
,
K
n
Q81=)6k<T =T k
,

n
R
X Tăng trưởng của vi khuẩn:
s t( )= Ae
rt
-0EA"#qs t( )"#q)t
r].6-"e(r > ,)t)6-"e
Ví dụ 1:'(") cp001(c0K)<0?`A1r0C;1&)
<p0?`A 1r08-91=)<"#p06?,"#p0-"3E(
6
A.K−0+) B. ) C.K−0K) D.K++)
Lời giải
[ET =T
,
,
t
ET (K) =T
,
,
K
HMt
,
)p0?` 1r/<T
,
,
t
,
=
T (K)
=
T
,
,
K
+ +
⇔,
t
,
=
+,
K
⇔ 0,
t
,
= 0
+
,
K

⇔ =t
,
K−0+Chọn A.
Ví dụ 2:$") 1(cp001(/"3N)t)p0>A 1r/78
-91=)"#cp06?,"#cp0-"3E(6C;1&
)<cp0?`A 1r/G
t ,
t
A. B.
J J
C.t −0J
t
D. 0J
Lời giải
[ET =T
,
,
t
HMt
,
)0 )@p0?`A 1r/&-T
,
,
t
,
=
T
=
T
,
,
t
J J
N&-,
t
,
=
,
t
⇒ = −t
,
t 0JChọn C.
J
Ví dụ 3:[-0.?p00T2"#T\1?2EE-02&-/'32&1(
E1c0MX.$1?c.-p00T2E@"#T\@8a-Ec
TUAA.1sTI=-#-I."7p00T2"# -D1r"3'(
") 1("#p00T281L5T.A1r/78-9:1(p0?c
-@J"#E(?c-@`p0e1M)@1"N0D&p0>
d?`A1r/G
A.P×0K+
J
B.J C.P× D.P×0KL
J
Lời giải
HMA"#p0@?`A1r/<"#p0 A
N"#p0JA&-n"#p0J
n
A
Y@"#p0?`A1r/<J
n
A =
,,
A⇒ =n0
J
,,
= 0K+
J
⇒)@p0?`A1r
L L
/t = P0
J
K+Chọn A.
lOMoARcPSD|40651217
Ví dụ 4:N"#`1(0q-0?lA.1"#AB0s t( ) = Ae
rt
-0EA"#qs t( )"#qEtQ?:Rr].6-"e
(r > ,)tQAB0?:R)6-"e78-9"#qE+,,0+
)E+,,0C;02@d:b"#qK+,,0G
A.J+) B.L+) C.K+) D.+)
Lời giải
[B0-E+,,= +,,e
+r
⇒ e
+r
= J
hE"#qK+,,0<
t t
K+,,= +,,e
rt
⇔ e
rt
= KLJ⇔(e
+r
)
+
= KJL⇔ J
+
= KLJ⇔ =t+0
J
KLJ= K+)Chọn C.
Ví dụ 5:N"#`1(0q-0?lA.1"#AB0s t( )= Ae
rt
-0EA"#qs t( )"#qEtQ?:Rr].6-"e
(r > ,)tQAB0?:R)6-"e78-9"#qE,,0
+)EJ,,0C;02@d:b"#q6?,03"#
G
A.t = ) B.t = ) C.t = ) D.t = )
Lời giải
[B0-EJ,,=,,e
+r
⇒ e
+r
= Jh
E"#q6?,
t t
,= e
rt
⇔ e
rt
=,⇔(e
+r
)
+
=,⇔ J
+
=,⇔ =t +0
J
,= )Chọn C.
Ví dụ 6:N_6-"e`1(0q"#AB0f x( ) = Ae
rx
-0EA
"#qr].6-"e(r > ,)xQAB0)R)6-"e78
qE,,,0,)+,,,0C;02<"#q6?,

A.+K,Q)R B.+,Q)R C.,0
+
,Q)R D.,0
+
K,Q)R
Lời giải
[B0E+,,,=,,,e
,r
⇒ =r
+
HMx
,
))@q6?,&-,A = Ae
,x
,
⇒ x
,
=,0
+
,Q)RChọn
C.
Ví dụ 7: [Đề thử nghiệm Bộ GD&ĐT 2017] N"#`0qZ-01(?lA.1
"#AB0s t( ) = s( ),K
t
-0Es( ),"#qZ:s t( )
"#qZEt?:78J?:<"#qZ4K+0C;02@d
:"#qZ,-.0G
A.LW?: B.O?: C.P?: D.K?:
Lời giải
[Es( )J= s( ),K
J
⇒ s( ),=
s
( )
J
= PWK+<0
W
j0Es t( ) =,-.0t,,,,<0,,,,= s( ),K
t
⇒ K
t
= =KW
⇒ =t 0KW
K
= P?:Chọn C.
Ví dụ 8:[-01-")&I")%&+&"#0qZ6
D?l,&"#0q76D?H *E,,0qZ
K,,0q7;0D&&-01-")E<"#09
8-9(6-"e`1=0e1M)@1"G
A.+×0K & B.+×0K &
C.,×0K & D.,×0K &
Lời giải
H *x&"#0q9hEE
x x x x x x
,,K
+
= K,,J
,
⇔ K
+
= KJ
,
⇔ K
+−
= J
,
⇔ − =
x
x
0J
K
⇔ x(K−0J
K
)=,⇔ =x
,
+ , K−0J
K
uEK−0J
K
= 0
K
L
=
⇒ =x
,
=,×0K
L
&Chọn D.
J 0K
L
K−0J
K J
J
lOMoARcPSD|40651217
Ví dụ 9:N"#`0-H-01(?lA.1"#AB0s t( ) = s( ),J
t
-0Es( ),"#-H:s t( )"#-HE)t?:78
+?:<"#-HW+,,,0C;02@d:"#-H
KK,,+,,,0G
A.W?: B.J,?: C.KP?: D.+?:
Lời giải
N+?:<"#-HW+,,,0&-W+,,,= s( ),J
+
⇒ s( ),= 0
HMt
,
?:)@EKK,,+,,,0-&-KK,,+,,,= J
t
,
⇒ =t
,
W?:Chọn
A.
BÀI TOÁN 5. TIỀN GỬI TIẾT KIỆM
H *1(")1=c*mQR-0ncN !-d
*`
[cm(+r)
n
[cm(+r)
n
−
vvvvvvvvvvv
[cn−m(+r)
N&-nc !"#T = m(+r)
n
+m(+r)
n
−

++m(+r)
u

= m(+r) = u
−q
n
= m(+r)(
+r
)
n
−
S?TU`?23
= + T
−
q
r

q r
Chú ý.$8c*M

c*M
K
vvcn−*
M
n
−
<T = M

(+r)
n
+ M
K
(+r)
n
−

++ M
n
−

(+r)
Ví dụ 1:7[1E1(-./+c<1=c? *020D
8`2,451=cQ1-l8FIR
A.4J+J,/ B.4++J,/ C.+W+J,/ D.4++L,/
Lời giải
[B0c8%?-D"#T = m(+r)
(
+r
)
n
−
r
⇒ m = T r n = ,,,,,,,45+ ≈ 4J+J,/Chọn A.
(+r) (  +r) −(+,45) (  +,45) −
Ví dụ 2:'(")c*02-./788.1`2
-0fc-<*9,J+5C;61")EE0D
A.,LJ-./ B.KKW-./
C.K+-./ D.L,-./
Lời giải
[B0c8%?-D"#T = m(+r) (
+r
)
n
−
= +(,J+5
)
(
+,J+5
)
K
−
r ,J+5
⇒ ≈T KKW-./Chọn C.
Ví dụ 3:'(")11(61? EK,-./@1aB1c&C;")E? *
021(0 "c0D788.1,KP5c
QM8f R
A.4L-./ B.PW-./ C.L-./ D.L+-./
Lời giải
[B0c8%?-D"#T = m(+r)
(
+r
)
n
−
r
⇒ m =
T r
n
=
K,,KP5
K
≈4JP-./Chọn A.
(+r) (  +r) −(+,KP5) (  +,KP5) −
BÀI TOÁN 6. TRẢ GÓP HÀNG THÁNG
H *1(")&T :1(c@d&&1=c")E- m
nc
lOMoARcPSD|40651217
N !-dTncT (+r)
n
N!-dm`cm(+r)
n
−
N!-dm`cm(+r)
n
−
K
N!-dm`cnm
$"%&- m(+r)
n−
+m(+r)
n−K
+ + = m
m
(
+r
)
n
−
r
N&-l? - T (+r)
n
−m
(
+r
)
n
−
r
Y@- 8#<T (+r)n −m (+r)n −= ,⇔ m = T r( +n r)n 
r (+r) −
Ví dụ 1:'(")&2,,-./3,P5cB0;%1=c
")E>- 02+-./- c"8088#Qc\E
@- T"3+-.RC;0Dc<")E- "#8#2
A.KK B.KJ C.KL D.K
Lời giải
=
Ar( +
n
r)
n
3a- cA&2r
[Ea
(+r) −
j0EE+=
,,,P5
(
+
n
,P5
)
n
⇒ =n K4KDKKc>- 8#Chọn A.(+,P5)
−
Ví dụ 2:Z7<11(8.0cO-./B0<- -"3J,5?l-
E?c3,O5c78-97<1- #*B0cN:1(
c@d&17<b- #- #D8?c:1(c- #
e1="C;Kc7<1- 8#<c7<? - 0*
0DQ1-l8/RG78&-0)7<- #
A.++4,,,/ B.PO+,,,/ C.4,L,,,/ D.WW,,,,/
Lời giải
N7<#`9OP,5= 4J-./
$#`7<3`nc"#AB0[ N
n
=T (+r)
n
−a (
+r
)
n
−
-0ETr l#a- E?crcn
c
[B0E /Chọn A.
Ví dụ 3:7Z11(81c&Ac,-./9<- E?3,P5c
Y@11c&T\Z- J-./h@dc8?B01Z- 1=c
+,,/C;c\Z? - 0D<8#Q1-l8FI/R
A.L,/ B.JP+/ C.JO/ D.LPK/
Lời giải
T r
(
+r
)
n
S?TUm
=
n
3m- 1=crT(+r) −
P,,,P( +,,,P)
n
? - N&- c
N&-? - c9(n−+,,,,,)≈ JO/Chọn C.
Ví dụ 4:'(M11w?0BPxEcK,-./X<EDB11y
1- E?<B0c3+51=c$8B11Wc- 8#<1=
cB1- mQ8f "#f&-l</R78-0)E"F`
1yB11=c9K+-.0cm3"F`1yEE
A.zFO+W,,,/ B.$FOK,,,/
C.zFPWO,,,/ D.$FKW,,,/
Lời giải
YrT = K,-./
T (+r)Wr T (+,,+)W,,+
[Em = W⇒ m = W≈P-./(+r) −
,+−
j0E- E?AFK+−P≈ ,PWO-./Chọn C.
lOMoARcPSD|40651217
BÀI TOÁN 7. MỘT SỐ DẠNG TOÁN KHÁC
Ví dụ 1:[B0T_c031DUT".&<-k"#T`"3Z>8
,,61k$"T0f AV1I1({&-(1@c%D@d61K-e1
DU6DL51=610361-"3C;0D61TT_-k`"3Z>
8G
A.JO B.L+ C.L D.LK
Lời giải
HMTDU1=61B0T_Ax N&-T_-kT,,x
HMt61_8DU8T&-x+ x(,L)+ x(,L)
K
+ +x(,L)
t
=,,x
⇔ x −(,L)
t+
=,,x ⇔ −(,L)
t+
=,,⇒ ≈t LK61Chọn D.
− ,L − ,L
Ví dụ 2: [Đề thi chuyên ĐH Vinh năm 2017] cA 2&.
A&D2`&81[-cYED[B0
|}jQ[C#?cxc-@883R.([-c
Y6D<c-I80 1$")"3A
-9.([-c6D1K°C<c-I80
 1J5~l.([-c6D1+°C<c-I8
0 1,578-98.([-c6D1t C° 
c-I80 1f t( )5<f t( ) = k a
t
-0Ek ac9T"F
h.([-c6D10D°C<c-I80 18K,5G
A.WL°C B.OJ°C C.P4°C D.4P°C
Lời giải
[B0E

k a
k a
+
K
=
=
,
J5
5
( )[<1t0
0k a
t
= K,5
[dQR⇒ =
k
J
a
5
K
a
J
=
,
J
⇒ =a
⇒
J
a
5
K
a
t
= K,5⇒ a
t−K
=
K,
J
⇒ − =t K
a
J
⇒ = +t K 0J,
K,
J
≈ 4PChọn D.
J
J
,
J
K,
0
Ví dụ 3:hccf1(1-")Q•"A"3"F1\vR")(>
 1TB0f")-&x B0I x( )= I e
,
−
µ
x
-0EI
,
")(`cc
b-&01-")µ.?`1-")E78-9"3@E.?
µ=L")A"#-9d(K1a8(2K,1<")(cc 1 ,
,
N&D02&3
A.W B.O C., D.O,
Lời giải
[Ex = K,− =KW1
[B0E,
,
= I xI( )
,
= eµx = eLW⇒ = eLW,≈ WWChọn B.
,
−
Ví dụ 4:'(.0??T"#?"#AB0Q t( )= Q
,
−e
KJt
3t
0 )A9)Q
,
T"#?Q?&R$8.0??d:
?QT"#?:b?,5R<02>?"#O,5Q8f 1-l8
?-61RG
A.t ≈+Lh B.t ≈Kh C.t ≈h D.t ≈JLh
Lời giải
[EQ t( ) = −e−KJt ⇔ O,= −e−KJt ⇔ e−KJt = ⇔ =t −K ≈+LhChọn A.
Q
,
,, , J ,
Ví dụ 5:^Zb131"Fe@1-./1(c61<Z
"#6"FL,5C;-lK,611"FZ%"#0DQ1-l8
k%??2T?q&RG
A.PK4PL-./ B.P4PL-./
C.W+WPK-./ D.P4WJP-./
Lời giải
NZ81"#-0J61JK= J4-./
NZE"#W611
S

= J4+J4( + L,5)

+ +J4( + L,5)
+
+J4( + L5)
4
NZ%K61Q61OK,RS
K
= KK( + L5)
4
j0EZ"#S -./Chọn D.
lOMoARcPSD|40651217
Ví dụ 6:'(/21•"3r@1OE-&21'")(21@1
McO1(0 R"#AeL
M
= 0
k
K
Q7BR3k978R @1O(
0•AB1")(21ABL
A
= JBenL
B
= +Ben[A1")(21-
@1ABQ1-l8kT?q&R
A.J+O7B B.J,47B C.J4O7B D.L7B
Lời giải
[E
LA = 0 OAk K= J⇒OA = ,,,k ⇒ AB = ,k + ,k = ,k
LA = 0 OBk K= +OB = ,,,,k ,, ,,, ,,,
HMN-@1AB⇒ ON =
AB
−OB =
,k
−
,k
=
O,k
K K,,, ,,, K,,,
k K,,,
K
k
N&-1")(21N9L
N
= 0ON
K
= 0
W,k
≈ J4OBenChọn C.
Ví dụ 7:'(@"3ETA,,,A$")1el0"3 &0@@"3
[-0)%"3 &0@A?:[-0c)8?B0%"3 &)
?)-"3C;0 )02<@&"3Q8f :R
A.JL) B.L4L) C.LL) D.J4L)
Lời giải
HMx+0 )@"3 &&@E
)Chọn C.
Ví dụ 8:08<c-`?Ea-TRa
KK4
4,K61Q1("#Ra
KK4
4,K
61?2`&<€l1(*RN_?2`&"#AB0S = Ae
rt
-0EA"#
?Ear€.?2`&61(r < ,)t)?2`&S"#l
)?2`&C;+gamRa
KK4
L,,,61?2`&>l0D1Q1-l8
Jk%??2RG
A.,WW4Q1R B.,KJQ1R C.,PO+Q1R D.,OKJQ1R
Lời giải
YD>Ar  A = Ae
4,Kr
⇔ =−r
K
K 4,K
[&A = +Q1Rt = L,,,r =− 0S = Ae
rt
<1"#S ≈ ,WW4Q1RChọn A.
Ví dụ 9:X31DU6`--•"T_I<"#6T_-k`0,,
&$"_8@d&-e"#DU6`--6D1L503&
-"3EC;"#6T_-k`--•_8€`00D&G
A.LK& B.L& C.JO& D.L,&
Lời giải
HMx"#6DU-01(&B0T_I&-6E,,x
$&K"#DU6x( +,,L)
vvv$&tx(+,,L)
t
−
hEEx+,L

x+,L
K
x+ +,L
t−
x =,,x3t&_8DU8"F_
− ,L
t
N&- =,,x ⇔ ≈t L&Chọn B.
− ,L
Ví dụ 10:S?APQ0911B`&2A.11CRB0P = Pe
,
kx
Q11CR-0Ex(0Q091VRP
,
= P4,Q11CRc?Ae1"3@
(x = ,)k.& 178-9e(0,,,1<c?A4PKPQ11CR[Ac?
`Ae(0J,,,1
A.+KP,4Q11CR B.+J,KJQ11CR C.+J,PJQ11CR D.+L+,Q11CR
Lời giải
‚(0,,,1E4PKP= P4,e
,,,k
⇒ e
,,,k
=
S?Ae(0J,,,1
J
P = P4,eJ,,,k = P4, e,,,k )J= P4,  4PKPP4, =  +KP,4Q11CRChọn A.
Ví dụ 11:'(r.XEc A<[)@1,hE:K0X 31=0
X "#3)nQ3n&DT"FR<&%?)@1EE{1(-K
n
0X
c[&DT0<`0XbD&{a0LE%?8C;:4,h
?:E0D0%XG
A.LOOK B.JPK C.OK4L D.+,,W
(
lOMoARcPSD|40651217
Lời giải
HMs
n
%"#-e)nEs
,
= K~s

= s
,
K= L~s
K
= s
K

+ s
,
K
K
=4
s
J
= s
K
K+ s
K
K
+ s
,
K
J
= 4L~s
L
= s
J
K+ s
K
K
K
+ s
K
J
+ s
,
K
L
= K+4s
+
= s
L
K+ s
J
K
K
+ s
K
K
J
+
s
K
L
= O4,~s
4
= s
+
K+ s
L
K
K
+ s
J
K
J
+ s
K
K
L
= JPK
hE%e)4T = s
J
+ + +s
L
s
+
s
4
= LOOK0Chọn A.
BÀI TẬP TỰ LUYỆN
Câu 1:'(")*,,-./01(23,L5c78-98-:
;2<1=c>%?0@A0c8?B0C;
:4c")E"#iQ R30T"32&8-0
0 )&")E-:-&G
A.,KLKL,,,/ B.,KLKJ,,,/
C.,K,4,,,/ D.,K,P,,,/
Câu 2:'(")*8.18,-./01(23P51(6178-98
-:-;2<1=61>"#%?0N+6113
-:<")E"#
A.L,K4-./ B.+,P-./ C.L,K4-./ D.J+-./
Câu 3:'(")*8.1023WL56161"#%?0
[A61@")E*@"#FK*
A.,61 B.O61 C.W61 D.61
Câu 4:'(")*2,,-./3<Jc+51(fgB0<
V?QJc>A(0RN:4c")E*D1+,-./3<
"-"3E[A")E%"#61QAd*DRG
A.PO4P4-./ B.PP4P4-./C.PW4P4-./
D.P44P4-./
Câu 5:Y@_.80T0ZqI1(&d2&)^E
+,,-./*8.13,L5cB0<V?N,cZ*D10
J,,-."ccE&,+5cC;K61@d:*
Z%"# !0DGQhA?%??2R
A.WPO4OJ4,, B.WW,LJW4L, C.WPO4OJ+, D.O,PKPWK
Câu 6:'(")*8.101(23456178-98-:-
;2<1=61>"#%?0@A0618?B0C;A
0D61")E"#Q *R?* I-0
0 )&&")E-:G
A.J61 B.,61 C.61 D.K61
Câu 7:^V*8.1K,,-./023<V?PK51(61
C;+61VQ !R302&G
A.KWJL+,,,/
B.KWJ++,,,/
C.KWJLK,,,/
D.KWJ+,,,/
Câu 8:j283"#T_0cB0P t( ) = aB
bt
-0Ea bc9t61A
T2[B0._8T28361O+,K+4,-.")~T28361OW,J,L,
-.")C&T_0cT28361K,K,G
A.JWKJ-. B.+J4,-. C.JO+L-. D.L,P-.
Câu 9:'(")*+,-./01(23P56178-98-:-
;2<1=61>"#%?0@A0618?B0C;:
+61")E13-:<")E%"#30T"32&G$8
-00 )&")&-:&
A.K,KW-./ B.P+-./ C.P,KW-./ D.4P+-./
Câu 10:'(")*+,-./01(23WL5611=61
>"#%?0@A0618?B0$")E>i"# !W,-.
/n61C;8-00 )&")E-:&<n
30T"32&
A.+ B.4 C.+ D.P
Câu 11:'(")*2,,-./B0<V?r = ,+51(cQ@d
cK"#AB0?-61E"#`c-"3E3`c-"3
ERNA0Dc")EEFK+-.
A.L+c B.L4c C.LPc D.LLc
Câu 12:'(")*8.13*Z/3451(618-98-:
-;2<1=61>"#%?0@A0618?B0
N,61")E-:-"#!F,,-./GC;")E? 
*Z90DG
A.L+,JP,+WJ/
B.++WJOLPP4O/
C.K4LL4+WO/
D.JK+4J+/
Câu 13:^Z*+-./02B0@V?613P4+561
H *&C;+61Z"# !0D-./G
A.+,,P4+()
+
-./ B.+
+ K,,P4+()
+
-./
C.+[ +,P4+]
+
-./ D.+( +,,P4+)
+
-./
lOMoARcPSD|40651217
Câu 14: '(")*4-./02B0@V?613
P+4561C;0D61")*>EAK-./d*Q *
&R
A.+61 B.,61 C.K61 D.W61
Câu 15:7c7<*63?A]/Yr80+61? E`-D<
1=61c7<*021(0 8.1"9c-I0
2&8`2P56161"#%?0
A.4K-./ B.4K+-./
C.4KK-./ D.4KJ-./
Câu 16:78-9=0-")MX 1=D(?1(0 :%?M+-.
/71y'8.1@1=c*1("02B0<V?
C;1=cM? *0DQ1-l8<R@Oc-:!<
"#+-./8.,+5c
A.+LK,,,/ B.+++,,,/ C.++4,,,/ D.+L,,,/
Câu 17:'(")*8.1B0@V?"'=c")&8.11(
X/-/*02B0c3,W5c[<1X@J61@d&*
D")EE+,,-./
A.X = L,JP4 B.X =
L,4JP
,,W− −,,,W
C.X = D.X = L,J44
,,W−
Câu 18:Zx:",,-./01(&B01V?3+51(61H *
61&C;J61`x:3c-I02&G
A.,L4-./
B.+K-./
C.+K-./
D.L4-./
Câu 19:'(")E,-./*023JcQfgJcR45
fgB0<V?QJc>A(0RN:Jc")E*D1K,
-./3<"%&C;61Ad*D")E%"#
8f 0G
A.J+-. B.JP-. C.J4-. D.JW-.
Câu 20:'(")*8.1+,-./01(23P51(6178-98
-:-;2<1=61"#%?0$8+61
13-:<")E"#
A.K,KW-./ B.P,KW-./
C.J+-./ D.+,P-./
Câu 21:^Z101ek0 K,,,,,,,,/0&KJK,Ke1(0 
614,+5C;Z"0D-D0 &0&KJK,,P@"#1UD
-G
A.LO,OO4+K+/
B.LO,OO4+K4/
C.LO,OO++K+/
D.LO,OW4+K+/
Câu 22:^Z*OW-./8.13WL561961"#%?0;
B0cE<0D61")E"#K,-./Q8-9&
R
A.O61 B.W61 C.P61 D.,61
Câu 23:^Z*8.13WL561961"#%?0C;0
D61")E"#?G
A.O61 B.W61 C.P61 D.,61
Câu 24:ZZ1-IcJ,,-./B0?"F- E?$81=cbdc
Z- ++,,,,,/I"- ,+5c<0Dc
Z- 8-D
A.n = 4L B.n = 4, C.n = 4+ D.n = 4L
Câu 25:7Z*,,-./00 IAV?3W561N+61-:
0(T\1(*@*l8?UB1*2-0+613\
[A"#,61
A.WLK-./ B.+WOK-./
C.O-./ D.PW-./
Câu 26:'(0"30TW,°F"#"01(1c&1cJK°F $.(`0Te?:
t"#AB0I%$Bƒ0eT t( )= JK+ LW,O( )
t
x 11c0T-00
2@.(+,°F G
A.t =+4?: B.t = OJ?: C.t =K?: D.t =L?:
Câu 27:")(1(-%(MQ-B-R"#0eM = 0A−0A
,
3AD(-
A
,
D(qQ9RY8€XX 1(-%(eN„-0E")(WJ(
-B-[-0\61E-%(ce$1'…ED(1F?L
[A")(`-%(e$1'…
A.WO(-B- B.JJK(-B- C.K,P+(-B- D.(-B-
lOMoARcPSD|40651217
Câu 28:H *n = f t( ) = n
,
K
t
"#c@-01(c1q)@1t)n
,
"#
c@:78(?c-@"#`q)@1tAf ′( )t H *1!
*En
,
=,,0qX%&(?c-@L)0D0qG
A.4,,0 B.,O0 C.+,,0 D.JK,,0
Câu 29:c02&a-0fc-<f#?>%"#1("#;0LQ1(/I`
0Rh1((?%`2&8<."#f#?`E†"E>%
D10Lku"#0L`(?%E>?2`&1(c%1?&@E
0L78-98MP t( )?-610Ll-01((?%`1(2&
-"edt61-"32&<P t( )"#AB0P t( )=,,,+( ) +P+,
t
( )5x2A1(
1!=d1(-<8-:")&"#0Ll-01!=E4+5$D
`-<8-:E302&G
A.LPP461 B.4J461 C.J+PL61 D.L,,,61
Câu 30:$6"#`1(-%("#A9E =PL,,
OLLM
3M(3B0c(
-B-[?Za &-1(-%(W(-B-6"#`E?L-%(a &
-?7C;E(3`-%(a &-?70DG
A.PK(-B- B.PW(-B- C.O4(-B- D.4O(-B-
Câu 31:$")f&"3x0xc-I`0
,
x [-0ci_…%x"#*TU
c@ 1c&A1&&f?CFk-00cM")*TU x@<1k
`1(&DT"F0EXATUZEnk<En =[A]+3[A]&D
3;F0r9ZC;K,P
K,P
E0DkG
A.OOOOk B.4444k C.444+k D.44OOk
Câu 32:}0Q}B-0Rq")-(2&D &UTkT(K,?:<
"#q}06?7€E4,q}0-0")-(C;W)
"#q}00DG
A.,,44JKO4,q B.K,WK+KP4,q
C.+W+OL4Oq D.JK+4,JP4Oq

LỜI GIẢI BÀI TẬP TỰ LUYỆN
Câu 1:S?TUV?&- !")EE4c
T =,,( +,L5)
4
≈,KLKL,,,/Chọn A.
Câu 2:S?TUV?&- !")E"#+61
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