Bài tập thể hiện phương pháp quy nạp, diễn dịch pps
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (149.63 KB, 14 trang )
!"#$%$&'()*+
,*(+*-.*/*0%*0'(1*#2"
1*$&31*4($%1* !1*5678#!
#!9):";<=->/
*<=-*?5@
Bài 1ABCD
∈
E
F
1#G.A
HIJIKI?IL>MN@O
>LIN@
H
>F@
N*/
PG.Q<Q'(#R83QQ1!S
$&'()654#11<=-(
H*T=-
DON*1A
HO
N>L5NIN@
H
<(U#V5
W-S"#X##VDO*.<
HIJIKI?I>LMN@O
>LIN@
H
>-R'()@
-.>F@#VDOIN*Y<
HIJIKI?I>LZN@I[L>IN@MN\O
[ ]
>IN@ L5>IN@ N
H
+
]A
HIJIKI?I>LZN@I[L>IN@MN\O
>IN@ L5>IN@ N
H
+
HIJIKI?I>LZN@IL>IH@O
>IN@>LI^@
H
5>FF@
9-R'()1A
>FF@
>LIN@
H
IL>IH@O
>IN@>LI^@
H
H H
L II_I^ L I^ILI^
H H
=
H H
L I`I^ L I`I^
H H
=
><(U#V@5
aD6
∈
E
F
<(U1A
HIJIKI?IL>MN@O
>LIN@
H
L*b
N
c
*BCD
∈
E
F
1#G.A
NILIJI?I>HMN@O
H
H
c
*BCD
∈
E
F
1#G.A
LIdIH`I?IL
O
N
H
>L
IN
ML@
eHABCD683Q
≥
H1]#G.
L
≥
LIN
N*/
aQ<Q'(#R83QEQ#!.
QS $&'()
H*T=-
fgOH1AL
H
Odh`><(U#V@
W-SQ#VDO
≥
H
.<1L
hLIN>-R'()@
-.Qi#VDOIN*.<
-.A
L
IN
hL>IN@INL
IN
h^IN
*9-R'()L
hLIN
L5L
h>LIN@5L
L
IN
hdILhLI^
L
IN
hLI^
bR<(AL
hLIN
L*b
N
c
*BCD683Q
≥
H1]#G.
H
hHIL
H
c
*BCD683Q
≥
L1]#G.
L
h
H
I^IJ5
eNABCR(
J
I
J
RJ1IR
J5
N*/A
P/<.3R1#X("5
1]X(#!.3R1!S
78(0X()S.
R=//S*1!S (Qj(<(
k<91!S #,<l&<9#,<lm9*21!S
$&'()?
(Q.9#1]1
1#X("(7*]R(IUR
Jn1>
J
I
J
@Z>M@O>
J
M@I>
J
M@URJ
(Q<)1
J
ZO>
^
MN@O>
H
MN@>
H
IN@
O>MN@>IN@>
H
IN@
O>MN@>IN@>
H
Z^IJ@
O>MN@>MN@>IN@>IH@IJ>MN@>IN@R
J5
a
J
ZiRJ
an.o$&-.
H*T=-
W-S1
J
I
J
M
J$IURJ
J
I
J
Z>I@URJ>N@
B>
J
I
J
@Z>M@O>
J
M@I>
J
M@
B21
J
ZO>
^
MN@O>
H
MN@>
H
IN@
O>MN@>IN@>
H
IN@
O>MN@>IN@>
H
Z^IJ@
O>MN@>MN@>IN@>IH@IJ>MN@>IN@RJ
$&3i1
J
ZRJ
EQ>
J
I
J
@Z>I@RJ
Oh>N@/((pD#X(
Oh-S<
aR(1
J
Z
J
RJIRJ5
L*b
e$&3A
N
c
*.oR(
`
M
`
R`1I
R`
H
c
*ER(
H
RJRJ5
BA-S
H
RJ$URJ
OhOJI>DON*H*L*^@
b#1
H
O>JI@
H
OHJ
H
INcI
H
1HJ
H
INcRJ5$
H
q,N*
^*d*N_>ON*H*L*^@Q
H
URJ5
EQ1
H
URJ/((/D-R<
H
RJ5
Oh#X(-S<
aR(
H
RJRJ5
eHABCD683**r*1
>
H
I
H
@>r
H
I
H
@
≥
>rI@
H
N*/
P/<.]#G.5
]A>
H
I
H
@>r
H
I
H
@M>rI@
H
O
H
r
H
I
H
r
H
I
H
H
I
H
H
M
H
r
H
M
H
r
H
IHr
O
H
H
I
H
H
IHrO>rI@
H
≥
c
Q1!S $&.#!.
]#G.Q5
H*T=-
W-SD683**r*1
>
H
I
H
@>r
H
I
H
@s>rI@
H
>
H
I
H
@>r
H
I
H
@M>rI@
H
sc
H
r
H
I
H
r
H
I
H
H
I
H
H
M
H
r
H
M
H
r
H
IHrsc
>rI@
H
scOhU<
Oh#X(-S<
a1>
H
I
H
@>r
H
I
H
@
≥
>rI@
H
L*b
N
c
*83***tIOH$
o]7]#G.(<#VA
H
≥
u
H
≥
H
c
*8$&**tON5BCII
≥
L5
!
ABCr
L
I
L
≥
r
H
Ir
H
∀
r
≥
c*
≥
cr
L
I
L
≥
r
H
Ir
H
N*/
P/<.]#G.t#X("5
]RR-:!(.1/S(<>rI@
4#11r
L
I
L
Mr
H
Mr
H
≥
c
>rI@>rM@
H
≥
c>F@
ar
≥
c*
≥
cQ]#G.#V*4#1(#X(-
.5
H*<=-
r
L
I
L
≥
r
H
Ir
H
r
L
I
L
Z>r
H
Ir
H
@
≥
c
>rI@>r
H
ZrI
H
@Zr>rI@
≥
c
>rI@>rM@
H
≥
c
ar
≥
c*
≥
cOhrI
≥
cOh>rI@>rM@
H
≥
c
∀
r
≥
c*
≥
c
ar
L
I
L
≥
r
H
Ir
H
L*
1$&3
N
c
***<#7L):5B>M@
H
s
H
H
c
*BCr
^
M
J
r
IrM
r
IN
≥
c
∀
r
≥
c
eHAB#G.(
>M@
>I@
O
5IN
5MN
N*/
P/<.]#G.
P!.]#G.Q#R#0R*
U.7<$%A>rI@OrZr
>rM@OrIr
(#1-Sp(#$%
5IN
5MN
4#11<=-(
H*T=-
1>rI@OrZr
>rM@OrIr
Oh
>M@ I
> @ Ma b
=
+
-Sp(#$%
> @
> @
a b
a b
−
+
O
5IN
5MN
L*b
$&3<=-:Q1(
.#G.(
N
c
*>I@>M@O
H
Z
H
O
H
Z
H
H
c
*>I@>M@O
H
Z
H
O
H
Z
H
L
c
*
H
NM NM
N H N
=
+ +
"#$
eNABC
∀
∈
C*hH
L
I^
sJ
>N@
N*/
1L*^*J<78AL
H
I^
H
OJ
H
H H
L ^
^ J
+
÷ ÷
ON
a8O
H
L
J
÷
O
H
^
J
÷
,R5
4#11<=-(
H*T=-
e]#G.>N@
H H
L ^
^ J
+
÷ ÷
sN
k
L
J
sNOhO
L
J
x
÷
,R
^
J
sNOhO
H
^
J
÷
,R
EQDhH1
L
J
a
÷
s
H
L
J
÷
u
^
J
a
÷
s
H
^
J
÷
a
L ^
^ J
+
÷ ÷
s
H H
L ^
^ J
+
÷ ÷
ON
L
I^
sJ
5
b
N
c
*e$&3A
B
∀
hH*
∈
C1_
IK
sNc
H
c
*e0'(
BC
∀
hH*
∈
Cr
I
sv
>r**v<78*Y<r
H
I
H
Ov
H
*r**v
∈
C@
eHAr
∈
[MH*c\5BCwrINw
≤
N
N*/
P/<.]#G.
P!.wrINw
≤
Nx.MN
≤
rIN
≤
N
4#11<=-(
H*T=-
1r
∈
[MH*c\OhMH
≤
r
≤
c
MHIN
≤
rIN
≤
cIN
MN
≤
rIN
≤
N
wrw
≤
N
L*b
e$&3
N
c
*r
∈
[MK*H\BCwr
H
ILw
≤
J
H
c
*r
∈
[MHL*L\BCwrINcw
≤
NL
#%&'()*%&
eN5
+Ar#,
∈
E#!H
MN
M
`
1) Phân tích
P/<8<Q'(#R3R
E]H
ZNOH
ZN
50'(]1
)83Q
Z
R>**<8
(Q@9,.9y1A
Z
O>Z@5z5aD*(Qz(Q*.<
Z
R
>Z@1!3#1#!
Z
R5
4#11<=-(A
H@ Lời giải
1!&$=%AOL*LIN*LIH
• DOL1
H
ZNOH
L
ZNOH
Lb
MN
b
O>H
L
MN@5zO`5z
M
`
OhOLH
ZNR`5
• aDOLIDON2OH
OhH
ZNOH
>LI@
ZNOH
>LI@
ZH
I>H
ZN@
OH
5>H
L
MN@I>H
MN@
kH
5>H
L
MN@R`
;H
ZNUR`QH
ZNUR`
OhOLIN2OLIHH
ZNUR`
bR<(ADOL>
∈
E@H
ZNR`
3) Khai thác bài toán
N
c
*J
ZNRd
H
c
*d
Z^
RJ
,%&
9p(AH
H
O^Q
^ H=
5(A
5aJ
H
OHJQ
555
OJ
5a`
O^dQ?O`
5aN
ON Q
N
O?
5a
H
H
555
L
=
÷
Q?O?
1) Phân tích:
P/<X{86:78u{8
6:78U/<8rA
H
r
=
9p(1AaH
H
O^Q
^ H=
5S ]n
-#$%51<=-(A
2) Lời giải:
5a
H
J HJ
=
Q
HJ J
=
5
5a
`
H
^d=
Q
^d `
=
5aN
H
ONQ
N N
=
5a
H
H ^
L d
=
÷
Q
^ H
d L
=
5
3) Khai thác bài toán
N
c
5a
H
^
555
J
=
÷
Q?O?
H
c
5a
H
NH 555
=
Q?ONH
N
c
5a
H
^ N_
J HJ
=
÷
Q
N_ ^
HJ J
=
H
c
5a
H
NH N^^
=
Q
N^^ NH
=
+#.(A
L H J L
H L H J L J
E NJ J ^ H
B L N
= + − − −
= + − + − + − +
5 (6#.Q5
5 EIBBIE
1) Phân tích
P/<#.7R*#!(6#.3"
1x"8:R1|8iD(5(#13
"g"8*n(#$%!(.(6x5
P!7>4@#.|RD(7>4@x"
8:R1|8i(5(#1*n(#$%!(
.x5
1<=-(A
Lời giải
5(6#.
1A
L H L
J L H
H L
EONJ IJ M^ MH
OM I>NJM^@ O>JMJ@ MH
OM INN MH
$&3A
H L H J L J
B L N `
= + − + − + − +
J L H
J
O>NI`@ I>NMN@ I>NMN@ MLIN
OK MLIN5
5 1A
J L J
EIBO>M INN MH@I>K MLIN@}
J L J
J L
J L
OM INN MHIK ILMN
O>NIK@ INN I>MHIL@IN
O` INN MJIN
J L J
EMBO>M INN MH@M>K MLIN@
J L J
J L
J L
OM INN MHMK ILMN
O>MNMK@ INN I>MHIL@MN
OMd INN IMN
2) Khai thác bài toán
N
c
5#.A
L ^ H
H L ^
N
>r@OMJr M IKr Ir
L
H
~>r@Or MJrMHr Ir M 5
L
H
c
5A
^ L
L H
^ H
>r@OHr MHr MrIN
~>r@OMr IJr I^r
C>r@OMHr Ir IJ
A
>r@I~>r@IC>r@
>r@M~>r@MC>r@
5
eLAhc*'hc5.oA
>IH@>'IH@>'I@•N_5'55<
x
→∞
-" !
MP/<).]#G.*1X(#!.
5
M]1)0:8U
/5a*n ]#G.U#!.]#G
.5
Me]#G.UAaD6hc*hc51A
I H 5
≥
1<=-(A
+-./
€ ]#G.U8U/*H51A
H H H
+ ≥
>N@
$&3*1A
' H H H'
+ ≥
>H@
' H '
+ ≥
>L@
E/R:>N@*>H@*>L@D(#$%A
>IH@>'IH@>I'@ H H5H H'5H '
≥
⇔
>IH@>'IH@>I'@ N_'
≥
5
k](or-O'OH5
a*]#G.#$%.5
0-1&*2
N
c
5.oA
H H H
H> @a b c ab bc ca
+ + ≤ + +
*D
* *a b c
<#7):75
H
c
5
cu cu cx y z
≥ ≥ ≥
A
N5x y z
+ + =
,<D]:!(.(A
M xy yz zx
= + +
5
34563789:;<=
{1/a"E1A•{/(
{t‚(#/A
•{/({t*
/(#.{{*
/(o{*
T /(*
e715
t1Q(/(#.*Q(/(o*Q(/(ƒ‚
>
&- " !
M P/<)-o<"$&5
M PXQ(x(8/(#.*8/(o*8/(54#/
n<#$%"$&]„5W-S6r<8/(
#.*<8/(o*v<8/(5
M 4U#…#$*nR<#$%8'("r*
*v$(A
I$DR#XR08/(<•{/(‚(A
rIIvONcc5
IPXR08t<•{t‚5#1*/(#.{
{*/(o{*/(715n1$&
A
JrILI
N
L
ONcc5
4#11<=-(A
5 T=-A
W68/(.<r*8/(o<*8/(<v>r**v<
8(Q$&t&Ncc@5
1"$&A
rIIvONcc
N
JrILI vONcc
L
P/<"$&]„*R(U#R#X(
":„"$&1U8">R(Sv#$%
7$&]„`rI^ONcc@5
(Q*r**v-<8(Q$&t&Ncc
Qq178()"* !#/1"A
N
N
N
r O^
ONK
v O`K
H
H
H
r O^
ONK
v O`K
L
L
L
r O^
ONK
v O`K
- 1&*2
$&31(A
eNA1/(X&5E.]*-/
(X#$%dLc5E.*/(X.]{
{(]NK†*/(X.{{(]NJ†Q-/
(X#$%NcKL5t.]‡/(X
Q(&5
eHA78185ER(RQ8Q
-8#1#$%8D<D&8#NddJ#&,58#
8RQ5
eLAB7UUHNdc)*#3L`51<)A_H
*_cJc58_HX(]#U8_c5t8
‡<)ƒ
+A0A•~(l‡'(-5‡'(-
<$=5B‡$=7R*{$=51$=-'(-U#:
#!5t1Q(ƒeQ('(lƒ‚
e<A
&- " !
M P/<)-o<"$&5
M k3"#!8<Q"„D
(5
M eQ(x(88'(l*5W68<r*8'(l<
51A rION`
M T)1‡'(-'(l*‡'(-<$=*‡
$=7R*{$=54#1*1$&A
NcrILONcc
4#1*#-"$&]„n#$%8
8'(l51<=-(A
*- ./
W6r<8x*<8'(lx>r*<8
(Q$&t&N`@5
91"$&A
rION`
NcrILONcc
>N@
>H@
4>N@(ArON`Z*>H@#$%A
Nc>N`M@ILONc
⇔
N`cM`ONc
⇒
ONc
⇒
rO`
a*1`'(-Nc'(-'(l5
- 1&
1$&3(A
eNAb86<D_ˆ^Kc9*#'(U
:#"UP51<)r9UUA<)Jc6<)^c
6*--<Ncr95t1Q(r9UU‡<)ƒ
eHA1`#X(NH95$R#!
U].NHxo(-ƒ
?@3;A;<=
A.oD6A
H
^
N
5
IN H
≤
&- " !
P/<).]#G.51! #,
YAz
≥
ezZe
≥
c#!.]#G.5
bR%DgR#0$&#$&n.#$%]
#G.#5
4#11<=-(A
*- ./
1A
( )
H H ^ H
^ ^
^
N H N > N@
c* 5
N H H> N@
H N
− − −
− = = − ≤ ∀
+ +
+
aA
H
^
N
* 5
N H
≤ ∀
+
k](or-q
H
N c N5
− = ⇔ = ±
- 1&*2
$&31(A
eNA.oD6hc*hc
H
5
N N
I
≤
eHA.oA
* * * ca b c d
∀ ≥
>I@>I@ I 5
≤
+AW-"<($&A
HrM OHMr5
>N@
&- " !
P/<)-"<($&1.](
,("#85P!-#$%xS](,("#8
$&oS ]:,("
#851!S ](A
c
c
5
B
A B
A
B
A B
≥
=
=
≥
= −
4#1*1<=-(A
*- ./
1A
r H
HMr c
H
Hr H r
r
>N@ 5L
H r c
r H
Hr r H
r H
≤
≥
+
− = −
=
⇔ ⇔
− ≥
≤
− = −
= −
I$=%NA
H
H ^5
L
+
≤ ⇔ ≤
I$=%HA
H H ^5
− ≤ ⇔ ≤
bR<(A
M aD
^A
≤
$&>N@1"
H
r
L
r H
+
=
= −
5
M aD
^A
>
$&>N@U"5
- 1&*2
eNAW-"<($&(98A
rMH O HrI 5
eHAf#,#!$&("(]A
r I NMr O
5
