!"#$!%& "$(!)*+) (+%*,-#.*#& /!% %*,+)&% 0$%,&/ /! #*!.#*&" .&,$)&%!" 1 2&,!23,*#&" /!(&),&2!.,+ /!(&),&2!.,+ /! 2&,!23,*#&" #$)"+ /! .*4!%*5. 6789 : 8" #&(;,$%+< . = 2 ! ) + " ) ! & % ! " /!>!) ?
(-
.*/0*1*%2* *% 3% )$&4& )* 5*%% 6& 76&1$8$7&7$9% )* 6:1 %;4*0:1 0*&6*1<
=-
>$ R *1 *6 7:%?3%2: )* 6:1 %;4*0:1 0*&6*1@ Q *1 *6 7:%?3%2: )* 6:1 %;4*0:1 0&7$:%&6*1@ I *1 *6 7:%?3%2: )* 6:1 %;4*0:1 $00&7$:%&6*1@ Z *1 *6 7:%?3%2: )* 6:1 %;4*0:1 *%2*0:1 A N *1 *6 7:%?3%2: )* 6:1 %;4*0:1 %&230&6*1@ $)*%2$8$B3* 6& /0:/:1$7$9% CDE>D< &-
( N ! Q ) ! R
F7)*-
Q ! I R =
Z ! Q N ! Z
N " (Q ! I ) .*1/3*12&G F-
6@8 )ABCADAEFGHIJE KAHILGM +-
HF$B3* *% 3%& 4$14& 0*72& %34I0$7& 6:1 1$#3$*%2*1 %;4*0:1 0&7$:%&6*1G &- 3.14 F7-
,-
4 5 3 2 1
)-
!
*-
!
5 2
HF$B3* *% 3%& 4$14& 0*72& %34I0$7& 6:1 1$#3$*%2*1 %;4*0:1 $00&7$:%&6*1G &-
J-
3
2 3
F-
2 3
7-
3
)*-
2!
2
! !
2 2
K*2*04$%* 1$ 6:1 1$#3$*%2*1 %;4*0:1 1:% 0&7$:%&6*1 : $00&7$:%&6*1G &F7)-
7.31 0.505005000…
!3.5478 5.070077000777 …
.*1/3*12&G &- .&7$:%&6@ F- L00&7$:%&6@ 7- .&7$:%&6@ )- L00&7$:%&6 !"#$%& ( )* +,
M-
O-
E& 134& N: 6& )$8*0*%7$&- )* 3% %;4*0: 0&7$:%&6 7:% 3% %;4*0: $00&7$:%&6 *1 :20: %;4*0: $00&7$:%&6< &- 5*0)&)*0: F- C&61: .*1/3*12&G &P6 /0:)372: )* 3% %;4*0: 0&7$:%&6@ )$8*0*%2* )* 7*0:@ /:0 3% %;4*0: $00&7$:%&6@ *1 :20: %;4*0: $00&7$:%&6< &- 5*0)&)*0: F- C&61: .*1/3*12&G &-
Q-
P6 7:7$*%2* )* 3% %;4*0: 0&7$:%&6@ )$8*0*%2* )* 7*0:@ *%20* 3% %;4*0: $00&7$:%&6@ *1 :20: %;4*0: 0&7$:%&6< &- 5*0)&)*0: F- C&61: .*1/3*12&G F-
R-
E& 80&77$9% 0*7S/0:7& )* 3% %;4*0: $00&7$:%&6 *1 :20: %;4*0: $00&7$:%&6< &- 5*0)&)*0: F- C&61: .*1/3*12&G &-
(T- .*&6$7* 6&1 :/*0&7$:%*1 4&2*4"2$7&1 %**71&0$&1 /&0& 0*/0*1*%2&0 7&)& %;4*0: *% %:2&7$9% )*7$4&6 7:4: 3% %;4*0: 0&7$:%&6G &- 5.26 F- 0.456 7-
3.1416
)-
145.235
.*1/3*12&G &-
526 100
@ F-
456 1000
@ 7-
10471 3333
@ )-
145090 999
6@6 +BACGHINEAD OIEGCIGD ((- K*8$%&G &- U/*0&7$9% F$%&0$&< F- !0:/$*)&) )* 7*00&)30&< 7- !0:/$*)&) 7:%432&2$V&< )- !0:/$*)&) &1:7$&2$V&< *- !0:/$*)&) )* *6*4*%2: %*320:< 8- !0:/$*)&) )* *6*4*%2: $%V*01:< (=- W:)& :/*0&7$9% F$%&0$& 734/6* 7:% 6& /0:/$*)&) )* 7*00&)30& A 6& /0:/$*)&) 7:%432&2$V&< &- 5*0)&)*0: F- C&61: .*1/3*12&G F(+- E&1 :/*0&7$:%*1 )* 3%$9% * $%2*01*77$9% *%20* 7:%?3%2:1 1:% :/*0&7$:%*1 F$%&0$&1< &- 5*0)&)*0: F- C&61: .*1/3*12&G &(,- P6 1*0 X34&%: & 20&VI1 )* 6& X$12:0$& X& 4&%*?&): )$8*0*%2*1 2$/:1 )* 0*/0*1*%2&7$9% %34I0$7&< L%)$B3*@ ?312$8$7&%): 13 0*1/3*12&@ 1$ 6&1 :/*0&7$:%*1 )* 134& A 0*12& )* 6:1 1$#3$*%2*1 1$12*4&1 %34I0$7:1 1* /3*)*% 7:%1$)*0&0 :/*0&7$:%*1 F$%&0$&1G &- >$12*4& )* %;4*0:1 0:4&%:1 F- >$12*4& F$%&0$: !"#$%& = )* +,
M-
O-
E& 134& N: 6& )$8*0*%7$&- )* 3% %;4*0: 0&7$:%&6 7:% 3% %;4*0: $00&7$:%&6 *1 :20: %;4*0: $00&7$:%&6< &- 5*0)&)*0: F- C&61: .*1/3*12&G &P6 /0:)372: )* 3% %;4*0: 0&7$:%&6@ )$8*0*%2* )* 7*0:@ /:0 3% %;4*0: $00&7$:%&6@ *1 :20: %;4*0: $00&7$:%&6< &- 5*0)&)*0: F- C&61: .*1/3*12&G &-
Q-
P6 7:7$*%2* )* 3% %;4*0: 0&7$:%&6@ )$8*0*%2* )* 7*0:@ *%20* 3% %;4*0: $00&7$:%&6@ *1 :20: %;4*0: 0&7$:%&6< &- 5*0)&)*0: F- C&61: .*1/3*12&G F-
R-
E& 80&77$9% 0*7S/0:7& )* 3% %;4*0: $00&7$:%&6 *1 :20: %;4*0: $00&7$:%&6< &- 5*0)&)*0: F- C&61: .*1/3*12&G &-
(T- .*&6$7* 6&1 :/*0&7$:%*1 4&2*4"2$7&1 %**71&0$&1 /&0& 0*/0*1*%2&0 7&)& %;4*0: *% %:2&7$9% )*7$4&6 7:4: 3% %;4*0: 0&7$:%&6G &- 5.26 F- 0.456 7-
3.1416
)-
145.235
.*1/3*12&G &-
526 100
@ F-
456 1000
@ 7-
10471 3333
@ )-
145090 999
6@6 +BACGHINEAD OIEGCIGD ((- K*8$%&G &- U/*0&7$9% F$%&0$&< F- !0:/$*)&) )* 7*00&)30&< 7- !0:/$*)&) 7:%432&2$V&< )- !0:/$*)&) &1:7$&2$V&< *- !0:/$*)&) )* *6*4*%2: %*320:< 8- !0:/$*)&) )* *6*4*%2: $%V*01:< (=- W:)& :/*0&7$9% F$%&0$& 734/6* 7:% 6& /0:/$*)&) )* 7*00&)30& A 6& /0:/$*)&) 7:%432&2$V&< &- 5*0)&)*0: F- C&61: .*1/3*12&G F(+- E&1 :/*0&7$:%*1 )* 3%$9% * $%2*01*77$9% *%20* 7:%?3%2:1 1:% :/*0&7$:%*1 F$%&0$&1< &- 5*0)&)*0: F- C&61: .*1/3*12&G &(,- P6 1*0 X34&%: & 20&VI1 )* 6& X$12:0$& X& 4&%*?&): )$8*0*%2*1 2$/:1 )* 0*/0*1*%2&7$9% %34I0$7&< L%)$B3*@ ?312$8$7&%): 13 0*1/3*12&@ 1$ 6&1 :/*0&7$:%*1 )* 134& A 0*12& )* 6:1 1$#3$*%2*1 1$12*4&1 %34I0$7:1 1* /3*)*% 7:%1$)*0&0 :/*0&7$:%*1 F$%&0$&1G &- >$12*4& )* %;4*0:1 0:4&%:1 F- >$12*4& F$%&0$: !"#$%& = )* +,
(J- Y0**@ )* 1*0 /:1$F6*@ 3%& :/*0&7$9% F$%&0$& B3* 734/6& 6& /0:/$*)&) 7:%432&2$V&@ /*0: %: 6& &1:7$&2$V&< (M- Y0**@ )* 1*0 /:1$F6*@ 3%& :/*0&7$9% F$%&0$& B3* 2*%#& *6*4*%2: %*320:@ /*0: %: 2*%#& *6*4*%2: $%V*01:< (O- Y0**@ )* 1*0 /:1$F6*@ 3%& :/*0&7$9% F$%&0$& B3* 2*%#& *6*4*%2: $%V*01:@ /*0: %: 2*%#& *6*4*%2: %*320:< (Q- K&): *6 7:%?3%2:
S
=
{1,2,3} 1:F0* *6 73&6 1* )*8$%* 6& :/*0&7$9% F$%&0$&
# /:0
4*)$:
)* 6& 1$#3$*%2* 2&F6&G #
1
2
3
1
1
2
3
2
2
1
2
3
3
2
1
L)*%2$8$B3* 6& /0:/:1$7$9% CDE>DG &-
!( 3# 3) # (1# 3) # " $
F-
! y " S # x " S
7-
!(1# 3) # ( 3 # 1) # " $
)*-
E& :/*0&7$9% F$%&0$& E& :/*0&7$9% F$%&0$&
=
(3# 1)
x # y x =
=
(1# 1) # #
*1 &1:7$&2$V&< *1 7:%432&2$V&< .*1/3*12&G )-
(R- >*& *6 7:%?3%2:
S
=
{a b c} 1:F0* *6 73&6 1* )*8$%* 6& :/*0&7$9% F$%&0$& ,
,
! /:0
4*)$: )*
6& 1$#3$*%2* 2&F6&G !
a
b
c
a
b
a
a
b
b
c
b
c
a
b
c
L)*%2$8$B3* 6& /0:/:1$7$9% 5P.KDKP.D< &- ( c!c ) "#( b!c ) !c$% =
F- E& :/*0&7$9% F$%&0$& ! *1 7:%432&2$V&< 7- ( a!a ) "#( b!c ) !a$% =
)*-
( b!b) "#(c!b) !c$% "#( a!b) ! ( ! )$% & ( c!b) =
a
c
.*1/3*12&G 7-
!"#$%& + )* +,
=T- >*& 6& :/*0&7$9% F$%&0$&
!
G
S ! S ! S
)*8$%$)& 1:F0* *6 7:%?3%2: S
=
{! ," ,# } 7:4: 1*
43*120& *% 6& 1$#3$*%2* 2&F6&G ! " # $
" # " #
# $ $ $
$ " " #
L)*%2$8$B3* 6& /0:/:1$7$9% 5P.KDKP.D< &- ! "# # " $ F- ! " # # " ! 7- ! "! # "# )- ! " ! # " # *- ! " # $ "! =
=
=
=
=
.*1/3*12&G )=(- K&): *6 7:%?3%2: S
=
{a,b,c} A ! 3%& :/*0&7$9% F$%&0$& )*8$%$)& *%
S )*
6& 1$#3$*%2*
4&%*0&G !
a
b
c
a
b
c
a
b
a
b
c
c
c
a
b
K*2*04$%* *6 V&6:0 )* V*0)&) )* 6&1 1$#3$*%2*1 /0:/:1$7$:%*1@ ?312$8$7&%): 13 0*1/3*12&< &- E& :/*0&7$9% ! *1 F$%&0$&< F- E& :/*0&7$9% *1 7:%432&2$V&< 7- E& :/*0&7$9% 2$*%* 3% *6*4*%2: %*320:< )- "#( a ! b) ! c$% c =
*- E& :/*0&7$9% %: *1 &1:7$&2$V&< .*1/3*12&G &- (@ F- T@ 7- T@ )- T@ *- ( ==- >$ ! *1 3%& :/*0&7$9% )*8$%$)& 1:F0* 6:1 %;4*0:1 *%2*0:1 7:4: a!b 2a " 3b < K*2*04$%* *6 V&6:0 )* V*0)&) )* 6&1 1$#3$*%2*1 /0:/:1$7$:%*1@ ?312$8$7&%): 13 0*1/3*12&< &- E& :/*0&7$9% ! *1 F$%&0$&< F- E& :/*0&7$9% ! *1 7:%432&2$V&< 7- E& :/*0&7$9% ! *1 &1:7$&2$V&< )- P6 *6*4*%2: %*320: )* 6& :/*0&7$9% ! *1 T< *- 4!3 3!5 .*1/3*12&G &- (@ F- T@ 7- T@ )- T@ *- T =
=
=+- >*& *6 7:%?3%2: ! a b " S <
S
=
Z
A 1*&
!
3%& :/*0&7$9% F$%&0$& 2&6 B3*
a *b
= a + b + 2 ab
@
,
P%2:%7*1@ !"2 * (5 * 1) #$ * !2 * (%2) * 0 # *1 $#3&6 &G " $
(
&- =@(+Q
F- =@+Q+
)
7- Z(@+,Q
)- Z(@=Q+
*- Z(@=+Q .*1/3*12&G *-
!"#$%& , )* +,
6@? +BACGHINEAD AEFCA EPLACND CAGMAD =,-
[ 5 &- 5*0)&)*0: 2
+
3
F- C&61: .*1/3*12&G F2
=J- >$% 32$6$\&0 6& 7&6736&):0&@ 0*13*6V&G
" 1 %" 3 % ! 0.5 1 ( ) $#1! 3 '&$#1 ! 5 '& 2
1!
0.1
5 9
.*1/3*12&G !8
=M- >$% 32$6$\&0 6& 7&6736&):0&@ 0*13*6V&G
(0.888... ! 1) ( 3.033...) 1 0.555...
+ 0.222...
.*1/3*12&G !
(3.333...)
=O- >$% 32$6$\&0 6& 7&6736&):0&@ 0*13*6V&G
(
)
0.5 ! 1
!2
3
1!
64 2
.*1/3*12&G
=Q- >$% 32$6$\&0 6& 7&6736&):0&@ 0*13*6V&G
1!
+
10
4
2 4 +
3
3
.*1/3*12&G
" ! ! $ (0.25) (0.5) ! (16) =R- D6 1$4/6$8$7&0 6& 1$#3$*%2* *]/0*1$9% (0.06666.....) $ 0.02222..... $ # 1
&-
1 12
F-
1 4
7-
3
)-
4
9
1
0.2
5.6 !
6
37
"1% $ ' ( 1.022...) ( 2.5) # 92 &
3
1
4 3
1
*-
73 5
% ' ' @ 1* :F2$*%*G ' &
12
.*1/3*12&G 7+T- L%)$B3* 73"6*1 1:% 6&1 /0:/$*)&)*1 B3* /:1** 6& :/*0&7$9% ^ )*8$%$)& 1:F0* 6:1 %;4*0:1 *%2*0:1G
a!b = a
2
+
b
!"#$%& J )* +,
+(- >*& 6& :/*0&7$9% F$%&0$&
!
)*8$%$)& 1:F0* 6:1 %;4*0:1 *%2*0:1 )* 6& 1$#3$*%2* 4&%*0&G a!b = a + b+ 2
P%2:%7*1 *1 CDE>U B3*G &- E& :/*0&7$9% ! *1 F$%&0$&< F- E& :/*0&7$9% ! *1 7:%432&2$V&< 7- _: 2$*%* *6*4*%2: $%V*01:< )- P6 *6*4*%2: %*320: *1 !2 < *- E& :/*0&7$9% ! *1 &1:7$&2$V&< .*1/3*12&G 7-
(
+=- >$ 1* 7:%:7* B3* &- T
a ! 0
#a " ! " b 0 ) ( ) %$ b
F- (
b +
= 2
a
& ( @ *6 V&6:0 )* '
7- =
)- (@TTT
( a ! b)
2000
*1 $#3&6 &G
*- =@TTT .*1/3*12&G &-
++- L)*%2$8$B3* 6& /0:/:1$7$9% 5P.KDKP.DG &- >$ F-
(
a
40
=
÷
4
)A(
b
+
2
*-
(
n
) @ *%2:%7*1 (ab ! ) <
(!4) (30 (!3)) ÷
3
7- >$ (!3) (!3) (!3) )- >$
9
=
0
+
y
)
5
@ *%2:%7*1
x
=
*1 $4/&0@ *%2:%7*1
x
÷ 2 =
2n
x >
0
2&4F$I% *1 $4/&0<
2
=
x+ y
.*1/3*12&G F+,- >*&
S
=
! A ! 3%&
:/*0&7$9% )*8$%$)& *%
S 2&6
a!b = b
a
+
B3*G
a
K*2*04$%* 1$ 6&1 1$#3$*%2*1 /0:/:1$7$:%*1 1:% V*0)&)*0&1 : 8&61&1@ ?312$8$7&%): 13 0*1/3*12&G &- E& :/*0&7$9% ! *1 F$%&0$&< F- E& :/*0&7$9% ! 734/6* 7:% 6& /0:/$*)&) 7:%432&2$V&< 7- E& :/*0&7$9% ! /:1** *6*4*%2: %*320:< )-
(1!2 ) !5
*-
E& :/*0&7$9% %: 734/6* 7:% 6& /0:/$*)&) )*6 *6*4*%2: $%V*01:< .*1/3*12&G &- (@ F- T@ 7- T@ )- T@ *- (
=
(
)
1! 2!5
+J- >:F0* *6 7:%?3%2: )* 6:1 %;4*0:1 %&230&6*1 1* )*8$%* 6& :/*0&7$9% F$%&0$&G
" $ a!b=# $% :/*0&7$9%G &- =Q
a
b
+
ab
,
b
a ,
si si
(a (a
+ +
) b) b
es
es
par
< !:0 6: 2&%2:@ *6 V&6:0 %34I0$7: )* 6&
impar
( 2 ! 1) ! (1 ! 3) @ *1 $#3&6 &G F- +T
7- +=
)- +M
*- ,T .*1/3*12&G 7-
!"#$%& M )* +,
+M- >*& 6& :/*0&7$9% F$%&0$& ^ )*8$%$)& *% &- Y&6736*
!@ a b
*
c d
ad
=
+
bc bd
( 1 2 * 1 3 )* 1 6
F- K*43*120* B3* 6& :/*0&7$9% *1 7:%432&2$V&< 7- K*2*04$%* 73"6 *1 13 *6*4*%2: %*320:< .*1/3*12&G &+O- >* )*8$%* 6& :/*0&7$9%
36 36 @
7-
0 1
*% *6 7:%?3%2: )* 6:1 %;4*0:1 %&230&6*1@ 2&6 B3*G
" $$ b=# $
%$a
a!b
a>b
a+b
a=b
b!a
a
,
,
P%2:%7*1 *1 CDE>U B3*G &- E& :/*0&7$9% *1 F$%&0$&< F- E& :/*0&7$9% 734/6* 7:% 6& /0:/$*)&) 7:%432&2$V&< 7- E& :/*0&7$9% %: 2$*%* *6*4*%2: %*320:< )-
(2 5)
*-
5
3
=
3 18
2
=
(5 3)
20
.*1/3*12&G )6@Q )AMGHIJE KA NCKAE +Q- K*8$%&G &- .*6&7$9% )* :0)*%< F- W0$7:2:4S& )* 6:1 %;4*0:1 0*&6*1< 7- !0:/$*)&) 0*86*]$V&< )- !0:/$*)&) 20&%1$2$V&< *- !0:/$*)&) &%2$1$4I20$7&<
+R- U0)*%* *% 8:04& &17*%)*%2* 6:1 1$#3$*%2*1 %;4*0:1G
e
,
!
2
,
5
,
2 ,
2
,T- `312$8$7&%): 13 0*1/3*12&@ 7&6$8$B3* *6 V&6:0 )* V*0)&) )* *12& /0:/:1$7$9%G aP%20* ):1 %;4*0:1 0&7$:%&6*1 1$*4/0* *1 /:1$F6* 2*%*0 :20: %;4*0: 0&7$:%&6*&% 6:1 %;4*0:1 0*&6*1 /:1$2$V:1 &- 5*0)&)*0:
x y < ,
>$
(
x
<
y
)
!1
@ *%2:%7*1 #
" x
<
1$
&
y %
F- C&61: .*1/3*12&G F-
,=-
!(a > b) ! (b > c)# % ( a > c) " $
!a b c " ! ,
,
&- 5*0)&)*0:
F- C&61: .*1/3*12&G &-
,+- >$*%): &-
2
x y
2
3
5
x y z
3% V&6:0 %*#&2$V:@ *6 /0:)372: B3* >LPc!.P P> _PdDWL5U *1G F-
xy
2
7-
yz
)-
2
x y
3
*-
2
x y
2
.*1/3*12&G 7!"#$%& O )* +,
,,- `312$8$7&%): 13 0*1/3*12&@ )*2*04$%* *6 V&6:0 )* V*0)&) )* 7&)& /0:/:1$7$9%G &- P6 7:%?3%2:
A
=
F- P6 7:%?3%2:
A
=
{ {
x / x
!!
+
x / x
!!
!
!
(3 (3
< x !
" !
4
)} *1 V&7S:< 2)} *1 3%$2&0$:<
< x ! "
.*1/3*12&G &- T@ F- ( ,J- >*&% 6:1 %;4*0:1 0*&6*1 a b c d x < L)*%2$8$B3* 6& /0:/:1$7$9% CDE>D< &- >$ a > b A c > d @ *%2:%7*1 b + d < a + c F- >$ a < b A c < 0 @ *%2:%7*1 ac < bc 7- >$ a < b A b < c @ *%2:%7*1 a < c )- a < x ! b *1 *B3$V&6*%2* & x ! (a b#" ,
,
,
,
,
.*1/3*12&G F,M- Y&6$8$B3* 7&)& /0:/:1$7$9% 7:4: V*0)&)*0& : 8&61&@ ?312$8$7&%): 13 0*1/3*12&< 1/4
&- 16
>
" 1 %! F- $1 ! 1.02' # 3 &
1/5
32
1
1
2
>
2
.*1/3*12&G &- T@ F- ( ,O- Si 0 < a < b y 0 < c < d entonces ac < bd &- 5*0)&)*0: F- C&61: ,
.*1/3*12&G &,Q-
Si a < b < c < d < 0, entonces ab > cd
&- 5*0)&)*0:
F- C&61: .*1/3*12&G &-
,R- L)*%2$8$B3* 6& /0:/:1$7$9% 5P.KDKP.D< &- !a
,
b c " ,
F- !a, b " 7- !a, b " )- !a
,
b c " ,
$( ab %
) # ( )&' & ab 0 # a 0 $ b 0 % a ) (( ) ( '( & ab 0 # a 0 $ b 0 % a ) (( ) ( '( &( a # b) $ ( a % c # b % c)( ' ) # {0}
*- !a, b "
=
bc
a
=
c
>
>
>
<
0$b > 0
))()
<
<
>
<
0$b < 0
))()
*$ '#1 1 1 ,& + ) = a + b/ ,+% a b ( /. .*1/3*12&G )-
{}
JT- >*&% 6:1 %;4*0:1 a, b, c ! Z ! 0 A 6&1 1$#3$*%2*1 7:%)$7$:%*1
a > 0 > b > c <
L)*%2$8$B3* 6& /0:/:1$7$9% 5P.KDKP.DG &F-
1
b+c
>
1
a
bc < a
!"#$%& Q )* +,
7)-
1
b
>
1
c
(b c)
2
+
>
1
a
2
*-
! 1 1$ # + & "b c%
<
1
a
.*1/3*12&G )6@R #NEHABFND GDNHIGKND G MND EPLACND AEFACND J(- K*8$%&G &- K$V$1:0< F- c;62$/6:< 7- _;4*0: /0$4:< )- _;4*0: 7:4/3*12:< *- c< Y< K< 8- 4< 7< 4< #- _;4*0: /&0< X- _;4*0: $4/&0< J=- P6 c
a!b
F-
a
+
b
(a
4
! b ) A
7- a 2 ! b 2
4
(a
2
)- a 2
! b ) *1G 2
+
b
2
*-
ab
.*1/3*12&G 7-
J+- W0*1 X*04&%&1 NB3* A& *12"% 7&1&)&1- V$1$2&% & 131 /&)0*1 7&)& =@ + A J )S&1 0*1/*72$V&4*%2*@ >$ 6& ;62$4& V*\ 1* *%7:%20&0:% /&0& 8*12*?&0 *6 ( )* *%*0: N)* 3% &e: B3* %: *1 F$1$*12:-@ *% 73"6*1 8*7X&1 )*6 /0$4*0 20$4*120* 1* V:6V$*0:% & *%7:%20&0 .*1/3*12&G +( )* *%*0:@ = )* 4&0\:< J,- H% 0*A 6* )& & 131 20*1 X$?:1 f MT@ f OJ A f RJ /&0& B3* 6:1 0*/&02&% & 6:1 %$e:1 )* 13 /3*F6:@ )* 2&6 4&%*0& B3* 2:):1 6:1 %$e:1 0*7$F&% *6 4$14: V&6:0< P%2:%7*1@ 6& 7&%2$)&) )* %$e:1 B3* 0*7$F$*0:% )$%*0: 83*G &- J F- (J 7- (R )- ,, *- ,M .*1/3*12&G *JJ- H% aK:% `3&%b 4:)*0%: 1* 7:4/0:4*2$9 & 1&6$0 7:% d$1*66* 7&)& + )S&1@ 7:% !&:6& 7&)& , )S&1@ 7:% c$0$&4 7&)& J )S&1 A 7:% >:%$& 7&)& M )S&1< P6 = )* 4&0\: )* *12* &e: 6* 7:00*1/:%)$9 734/6$0 7:% 2:)&1@ /*0: 13/*09 *6 /*07&%7*< L%)$B3* 73"%): 1*0" 6& /09]$4& V*\ B3* 6* 137*)*0" *12* $%7:%V*%$*%2*< .*1/3*12&G ( )* 4&A:< JM- Y&)& J X:0&1 &2*00$\& 3% V3*6: )* D5LD_YD@ 7&)& , X:0&1 3% V3*6: )* ED_ A 7&)& = X:0&1 3% V3*6: )* WDcP< >$ 6&1 20*1 &*0:6S%*&1 &2*00$\&0:% & 6&1 TMgTT )* X:A@ I12&1 V:6V$*0:% & 7:$%7$)$0 4&e&%& & 6&1G &- T(gTT F- T=gTT 7- T+gTT )- T,gTT *- TJgTT .*1/3*12&G F-
!"#$%& R )* +,
JO- !&F6: A c&0S& V&% & 7:00*0 *% 3%& /$12& &26I2$7&@ I6 6: X&7* *% = 4$%32:1 A *66& 1* )*4:0& + 4$%32:1< >$ /&02*% &6 4$14: 2$*4/: A X&% 7:00$): )30&%2* 3%& X:0&@ 6& 7&%2$)&) )* V*7*1 B3* 1* X&% *%7:%20&):@ &6 7:00*0@ *1G &- M F- Q 7- (T )- (= *- (M .*1/3*12&G 7JQ- H% 8&0: 1* *%7$*%)* 7&)& (= 1*#3%):1@ :20: 8&0: 7&)& (Q 1*#3%):1 A 3% 2*07*0: 7&)& 4$%32:< >$ & 6&1 (Qg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h= 6$20:@ MJT 4$6$6$20:1 A ,h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i&06& 2$*%* 3% 0*6:? B3* )& 3%& 1*e&6 7&)& MT 4$%32:1@ :20: 0*6:? B3* )& 3%& 1*e&6 7&)& (JT 4$%32:1 A 3% 2*07*0: B3* )& 3%& 1*e&6 7&)& +MT 4$%32:1< >$ & 6&1 RgTT )*6 )S& )* X:A 6:1 20*1 0*6:?*1 X&% 7:$%7$)$): *% )&0 6& 1*e&6@ )*2*04$%* 6& X:0& A *6 )S& & 6& B3* V:6V*0"% & )&0 6& 1*e&6 6:1 20*1 ?3%2:1< .*1/3*12&G c&e&%& & 6&1 (JgTT< M+- >&0& 2$*%* *% 13 2$*%)& 6:1 F:2:%*1 4*2$):1 *% F:61$2&1< P% 6& 7&?& D 2$*%* F:61$2&1 )* =, F:2:%*1 7&)& 3%& A %: 1:F0& F:29% &6#3%:< P% 6& 7&?& j 2$*%* F:61$2&1 )* =T F:2:%*1 7&)& 3%& A 2&4/:7: 1:F0& F:29% &6#3%:< P6 %;4*0: )* F:2:%*1 B3* X&A *% 6& 7&?& D *1 $#3&6 &6 B3* X&A *% 6& 7&?& j< kY3"%2:1 F:2:%*1 7:4: 4S%$4: X&A *% 7&)& 7&?&l .*1/3*12&G D [ J@ j [ M M,- H% 7&4/: 0*72&%#36&0 )* +MT 4 )* 6&0#: A (JT 4 )* &%7X:@ *12" )$V$)$): *% /&07*6&1 73&)0&)&1 $)I%2$7&1< P6 "0*& )* 7&)& 3%& )* *12&1 /&07*6&1 73&)0&)&1 *1 6& 4&A:0 /:1$F6*< kY3"6 *1 6& 6:%#$23) )*6 6&): )* 7&)& /&07*6& 73&)0&)&l .*1/3*12&G +T 4< MJ- H% *F&%$12& B3$*0* 7:02&0 3%& /6&%7X& )* 4&)*0& )* =JM 74 )* 6&0#: A RM 74 )* &%7X:@ *% 73&)0&):1 6: 4"1 #0&%)*1 /:1$F6*< &- kY3"6 )*F* 1*0 6& 6:%#$23) )*6 6&): )* 7&)& 73&)0&):l F- kY3"%2:1 73&)0&):1 1* :F2$*%*% )* 6& /6&%7X& )* 4&)*0&l .*1/3*12&G &- += 74@ F- =, 73&)0&):1< !"#$%& (T )* +,
MM- H% ?:V*% V$1$2& >&6$%&1 7&)& (Q )S&1@ :20: V& & >&6$%&1 7&)& (J )S&1 A 3% 2*07*0: V& & >&6$%&1 7&)& Q )S&1< g:A@ (T )* *%*0:@ X&% 7:$%7$)$): *% >&6$%&1 6:1 20*1 ?9V*%*1< kK*%20: )* 73"%2:1 )S&1 7:4: 4S%$4: V:6V*0"% & 7:$%7$)$0 *% >&6$%&1l .*1/3*12&G +MT )S&1< MO- P% 3% &64&7I% 1* 2$*%*% (TT 7&02:%*1 )* 6*7X* *%2*0&@ MT 7&02:%*1 )* 6*7X* 1*4$)*170*4&)& A ,T 7&02:%*1 )* 6*7X* )*170*4&)&< >* 0*B3$*0* #3&0)&06:1 *% 7&?&1 B3* 2*%#&% *6 4$14: %;4*0: )* 7&02:%*1< E& 4S%$4& 7&%2$)&) )* 7&?&1 B3* 1* %*7*1$2&% /&0& #3&0)&0 2:):1 6:1 7&02:%*1 *1 $#3&6 &G &- , F- Q 7- (T )- =T *- ,T .*1/3*12&G 7MQ- W0*1 /*00:1 )* 7&00*0&1 &00&%7&% ?3%2:1 *% 3%& /$12& 7$0736&0< >$ *6 /0$4*0: 2&0)& Q 1*#3%):1 *% )&0 3%& V3*62& & 6& /$12&@ *6 1*#3%): (( 1*#3%):1 A *6 2*07*0: (, 1*#3%):1< &- kY3"%2:1 1*#3%):1 )*F*0"% 20&%17300$0 /&0& B3* 6:1 /*00:1 /&1*% ?3%2:1 /:0 6& 6S%*& )* 1&6$)&l@ F- kY3"%2&1 V3*62&1 X&F0" )&): 7&)& /*00: *% *1* 2$*4/:l .*1/3*12&G &- M(M 1*#m F- ( *0 /*00:G OO V3*62&1@ =): /*00:G JM V3*62&1@ +*0 /*00:G ,, V3*62&1 MR- H%& 7&4/&%& )* 3%& $#6*1$& *% *6 7*%20: )* 6& 7$3)&) 13*%& 7&)& , X:0&1@ 7*07& )* I12& 1* *%73*%20& 3%& *12&7$9% )* F:4F*0:1 6& 73&6 X&7* 1:%&0 6& 1$0*%& 7&)& J X:0&1< D ):1 73&)0&1 )* 6& *12&7$9% )* F:4F*0:1 1* *%73*%20& :20& $#6*1$& B3* X&7* 1:%&0 13 7&4/&%& 7&)& = X:0&1< >$ & 6&1 TTgTT )* 3% 63%*1 1:%&0:% 6&1 7&4/&%&1 A 6& 1$0*%& ?3%2&1@ 6:1 )S&1 )* 6& 1*4&%& *% B3* 1:%&0:% 7&4/&%&1 A 1$0*%& ?3%2&1 4"1 )* 3%& V*\ 1:%G &- E3%*1 A ):4$%#:< F- E3%*1 A 1"F&):< 7- c$I07:6*1 A 1"F&):< )- c&02*1 A V$*0%*1< *- c&02*1 A ):4$%#:< .*1/3*12&G FOT- H2$6$\&%): *6 cnWUKU KP KPcU>W.DYLo_ KL.PYWD@ )*43*120* 6& 1$#3$*%2* /0:/$*)&) )* 6:1 %;4*0:1 *%2*0:1G Si m es impar, entonces m3 es impar.
!m " Z
O(- H2$6$\&%): *6 cnWUKU !U. YU_W.D..PYp!.UYD @ )*43*120* 6& 1$#3$*%2* /0:/$*)&) )* 6:1 %;4*0:1 *%2*0:1G !m " Z
2
Si m es par, entonces m es par.
6@9 !SBCADINEAD GMTAOCGIHGD O=- E& *]/0*1$9% &6#*F0&$7& *6 V&6:0 )*
(a
+
b
&- Z(M
(
x
2
+ 8 x
) 1* 6& /3*)* *170$F$0 )* 6& 8:04& ( x a ) +
2
! b < P%2:%7*1@
) *1 $#3&6 &G F- Z(=
7- (=
)- (M
*- =T .*1/3*12&G *-
O+- E& *]/0*1$9%G &-
a
1 8
3
a
3 4
a
a
@ *1 $#3&6 &G
5
F-
a
15 4
7-
a
6
)-
a
3
*-
a
4
.*1/3*12&G F!"#$%& (( )* +,
! O,- D6 )*1&00:66&0 6& *]/0*1$9% &6#*F0&$7&G # p " &- p ! q
2
3
1 3
3
!q
$ & @ 1* :F2$*%*G %
2
4
2
F-
p q ! 3 p
7-
p
)-
p
*-
p
2 3
4
2
! 3 p
2
! 3 p
2
! 3 p
2 3
q 1
3
q
4
q
4
! 3 p
3
+ 3 p
q
3
+ 3 p
1 3
! pq
3
2 3
3
q
2
2 3
q
2 3
1 3
2 3
+ 3 p
!q
2 3
3
q
2
!q
2 3
q
1 3
!q
3
.*1/3*12&G )-
OJ- D6 )*1/*?&0 6& V&0$&F6* &-
w
=
F-
w
=
7-
w
=
( y ! 2)
w
)* 6& *]/0*1$9% y ! 2
=
n
t ! z
1* :F2$*%*G
w
n
t ! z
(t ! z )( y ! 2)
n
t ! z
( y ! 2)
n
n
)-
w
=
" t ! z % $# y ! 2 ' &
*-
w
=
"( t ! z )( y ! 2 ) $ # %
n
.*1/3*12&G 7-
" OM- D6 1$4/6$8$7&0 6& *]/0*1$9% $ $ # &-
(3
x
2
! 4 y 2
3 x 3
!
y
2
%" '$ '$
3 x 3
+
y
2
%" x y % '$ + ' 1* :F2$*%*G '# 3 4 & &
)
2
F-
" x y % $ ! ' #3 4& " x y % $ ! ' # 9 16 & " x y % ( x ! y) $# 3 + 4 '& 2
7)-
*-
" $ $ #
2
2
3 x 3
% " x y % y ' $ + ' ! 4 ' & #3 4& .*1/3*12&G 7-
!"#$%& (= )* +,
"a!b% OO- D6 1$4/6$8$7&0 6& *]/0*1$9% &6#*F0&$7&G $ ' #a+b& a
&-
F-
b
2
a
c
7-
c
2
a c + 2 abc + b c 2
! 2 ab + b
@ 1* :F2$*%*G
2
a
)-
c
*-
2
c
c
.*1/3*12&G F-
"
(
OQ- D6 1$4/6$8$7&0 6& *]/0*1$9% &6#*F0&$7& $ #
4
!
x
y
)(
4
x
+
)(
y
x
+
y
)(
x
2
+ xy
2
4
+ y
)%'& 1*
:F2$*%*G
( (
&-
x
F-
x +
7-
x
)-
x
*-
x
3
3
2
( y ) (
! y )
x
x
2
2
+ xy
+ xy
2
2
+ y
+ y
2
2
) )
! y6 + y
!y
6
4
.*1/3*12&G 7OR- >$ 1* 7:%:7* B3*
(
x
2
+
y
2
2
+ z
"
(
%34I0$7: )* 6& *]/0*1$9% $ x ! y # &- T
50
=
F- (
) A B3* (49 ! xy !
2
) ( +
! z )
2
+
x
(
2
xz
! yz
=
0
) @ *%2:%7*1 *6 V&6:0
%
) '& @ *1 $#3&6 &G
y ! z
7- =
)- ,Q
*- JT .*1/3*12&G 7-
2
QT- D6 1$4/6$8$7&0 6& *]/0*1$9% &6#*F0&$7&G
x
! x 3 ! 2 @ 1* :F2$*%*G x ! 8
2
1
1
3
3
3
3
+1
2
1
3
3
+
1
1
x
&-
x
3
4 x
+
x
F4
x
!1
2
1
3
3
+
2 x
7-
x
x
2
Q(- D6 1$4/6$8$7&0 6& *]/0*1$9% &6#*F0&$7&G
x y
!1
2
x
÷
2
1
3
3
+
3
+1
2 x
+
x
*-
!1
2
4
x
3
1 3
+ 2 x
+
4
.*1/3*12&G )-
(" x m *$$ *)# y 2
m
x
)-
1 3
!4
!2
1
2
" y m ! $ x # 2
÷
2
% " xy ''' $ && # m
xm % y
2
÷
2
÷
+ '&-,
ym % x
2
1* :F2$*%*G 3
&-
xy m
2
F-
m
( xy)
2
! xy $ 7- # & "m%
m
)-
2
3
2
x y
*-
2
x y m
2
3
.*1/3*12&G 7Q=- >$4/6$8$B3* 6& *]/0*1$9% &6#*F0&$7&G
x
6
+ x
x
9
3
y
3
!y
+ 9
y
6
(x
6
! y6
) .*1/3*12&G
x
3
+
y
3
!"#$%& (+ )* +,
x x + x
Q+- >$4/6$8$B3* 6& *]/0*1$9% &6#*F0&$7&G
!1 x
x
!
x
!1 x
.*1/3*12&G
!2
x
Q,- >$4/6$8$B3* 6& *]/0*1$9% &6#*F0&$7&G
! x x
2
2
x
+1
+ x
4
+1
4
!1 1
.*1/3*12&G x
! 2a QJ- >$4/6$8$B3* 6& *]/0*1$9% &6#*F0&$7&G # " 2a
2
2
$! 2a + 9 a + 9 % " 4a
2
+ 7a + 6
2
2
$ & + 9a + 2 %
x
2
9 +
+ 10 x + 24
!1
+ 17a + 8
( (
.*1/3*12&G
QM- >$4/6$8$B3* 6& *]/0*1$9% &6#*F0&$7&G
2
18 ! 3 x ! x
2
) (2 3) ( 4
) 1)
a +8
a +1
a+
a+
4 x ! 5
+ x
2
+ x
! 12 4
.*1/3*12&G
x +
x ! 1
QO- >$4/6$8$B3* 6& *]/0*1$9% &6#*F0&$7&G
x + 1
(2
x !
!
)( 4
3
x
(
x
2
x + 1 x ! 1 2
!
+ 6 x + 9
)
"
( 27
!
8 x
3
)
)
1
.*1/3*12&G 2 1! a
n! 2 a
b+ n+1
b! m+3
.*1/3*12&G
" x + y $ x ! y >$4/6$8$B3* 6& *]/0*1$9% &6#*F0&$7&G $ $ 1! 1 $# x y 2
QR-
4 x
( 2 x )( x ( ) )( x ) >$4/6$8$B3* 6& *]/0*1$9% &6#*F0&$7&G ( x )( x ) p! n
QQ-
4
2
2 x
p+ m!n
% ' " x ! y % ' $ ' ' # x + y & ' & 2
2
3
3
.*1/3*12&G x
" 2 x ! 5 x ! 3 % (" x ' *$ RT- >$4/6$8$B3* 6& *]/0*1$9% &6#*F0&$7&G $ # x ! 9 & *)# 2
2
2
+ 6 x + 9
1 + 2 x
% " '÷$ & # x
! 9 %+ '+ 4 x + 3 &,
x 2
2
.*1/3*12&G
(
)(
x + 3 x
)
x +1
!3
!"#$%& (, )* +,
6 x + 12
x + 1
! x
R(- >$4/6$8$B3* 6& *]/0*1$9% &6#*F0&$7&G
!4+
x
x +
2
!5 11 x ! 22
!2
x
x + 7
.*1/3*12&G
1
R=- D6 1$4/6$8$7&0 6& *]/0*1$9% &6#*F0&$7&G
" $ x $ $ # 1* :F2$*%*G &- 1
2
F-
3
! ( xy ) x
2
1
% ' ' '" & $ x #
2 3
!y
+
3
y
2
7-
x ! y
1
+
y
2
1
1 3
" 2 x ! 3 xy + y % $ '@ ! ! x y 2 % # & ' & 2
1 3
)- x + y
x
*-
2 x
.*1/3*12&G &R+- E& *]/0*1$9% &6#*F0&$7&G !1
"8% !$ ' # 27 & ! " 1% $1 ! ' # 3&
3
" m ! 2mn + n +$ # m !n 2
2
2
2
2
%" m + n % '$ ' m ! n &
*1 $#3&6 &G &- !
3
F- !
2
2
1
7- !
3
)-
3
1
*-
3
2 3
.*1/3*12&G )-
" a ! x + ax a ! x % ( ) ' 1* :F2$*%*G R,- D6 1$4/6$8$7&0 6& *]/0*1$9% &6#*F0&$7& $ $ ' 4 ( a ! x ) # & 3
3
2
&-
(
a + x
)
2
2
F-
4
a + x
a
7-
4
! x
)-
4
a
2
+ x
4
2
*-
a
2
! x
2
4
.*1/3*12&G FRJ- K*8$%& .DYLU_DELqDYLo_< RM- .&7$:%&6$7*G 1
&2
7-
1 x !
*-
3!
+
x +
x !
F5
)3
3
3
y y y
1 3
n
1 4
a
4
! 3
8-
3
m +
3
9
9
+
b
+ 3
6
3
4
+
3
4 !"#$%& (J )* +,
RO- >$
3!
&- !36a
2
=
a
6 6
@ &6 0&7$:%&6$\&0 6& 1$#3$*%2* *]/0*1$9%
2 3
F- !6 a
7-
)-
0
6a
+3
@ 1* :F2$*%*G 2
*-
36 a
.*1/3*12&G ))ATMG KA FCAD DILBMA U CATMG KA FCAD HNLBVADFG RQ- `:1I 20&F&?& 6:1 1"F&):1 7:02&%): *6 7I1/*) & 131 V*7$%:1< >&F$*%): B3* 20&F&?& 2:):1 6:1 1"F&):1 6&1 4$14&1 X:0&1 A B3* /:0 7&)& M )S&1 7:F0& f (JT@ k73"%2: 7:F0& `:1I /:0 (J )S&1 )* 20&F&?:l .*1/3*12&G f+OJ RR- (TT 6$20:1 )* &7*$2* 73*12&% f (QR@ k73"%2: 73*12&% (=J 6$20:1 )*6 4$14: /0:)372:l .*1/3*12&G f=+M<=J (TT- H% &32:49V$6 0*7:00* +T r4 *% s )* X:0&@ k73"%2:1 r$694*20:1 0*7:00*0" *% ( t X:0&1l .*1/3*12&G (QTr4< (T(- >$ *6 =Ju )* 3%& 7&%2$)&) *1 MQ@ k73"%2: *1 *6 ,+u )* *1& 4$14& 7&%2$)&)l .*1/3*12&G ((M$ 3% %$e: 7&4$%& + r4 *% 3%& X:0& A 73&02:@ k73"%2:1 r$694*20:1 0*7:00*0" *% + X:0&1l .*1/3*12&G O<=r4< (T+- P% 3%& *173*6& X&A ,MO &634%:1 A *6 )S& )* X:A 8&62&0:% M+@ kB3I /:07*%2&?* )* &634%:1 *123V: &31*%2*l .*1/3*12&G (+<,u (T,- H% 20&F&?&):0 #&%& /:0 ?:0%&)& )* Q X:0&1 f(=,$ 13 ?:0%&)& &34*%2& *% =
10
m
3
)30&%2*
5 X:0&1<
!&0& 66*%&01* :20: )*/91$2: )*
3
7:% ):1 #0$8:1 $#3&6*1 & 6:1 &%2*0$:0*1@ *6 2$*4/: B3* 2&0)&0"@ *% X:0&1@ *1 $#3&6 &G &- M F- Q 7- (T )- (= *- (M .*1/3*12&G &8m
(TQ- JT X:4F0*1 2$*%*% /0:V$1$:%*1 /&0& =T )S&1 7:%134$*%): + 0&7$:%*1 )$&0$&1< >$ 6&1 0&7$:%*1 1* )$14$%3A*% *% (h+ A X&A (T X:4F0*1 4"1@ k73"%2:1 )S&1 )30&0"% 6:1 VSV*0*1l .*1/3*12&G =J )S&1<
!"#$%& (M )* +,
(TR- P% 3% /6&% )* V$V$*%)& *% d3&A&B3$6@ 3%& *4/0*1& 7:%120372:0& X& 7:%20&2&): JT :F0*0:1 B3* 8&F0$7&% , 7&1&1 )$&0$&1 20&F&?&%): Q X:0&1 )30&%2* 7&)& )S&< >$ 6& )*4&%)& )* 7&1&1 &34*%29 *% =Ju A 6&1 X:0&1 )* 20&F&?: )$&0$: )$14$%3A*% *% JTu@ k73"%2:1 :F0*0:1 &)$7$:%&6*1 )*F* 7:%20&2&0 6& *4/0*1& /&0& 73F0$0 6& %3*V& )*4&%)&l .*1/3*12&G OJ :F0*0:1< ((T- H%& F&0)& 7:%1203$)& 7:% +TT 2&F$B3*1 2$*%* 3% 6&0#: )* J 4*20:1 A 3%& &6230& )* + 4*20:1< kv3I 6&0#: 2*%)0S& 6& F&0)& 1$ 1* 7:%2&0&% QJT 2&F$B3*1 A 23V$*0& =
1 6
4"1 )* 6& :F0& *%
k + 60
k
n
:F0*0:1
)S&1< >$ *12:1 ;62$4:1 2*04$%&% 6: B3*
)S&1@ k73"6 *1 *6 V&6:0 )*
k
5n
l .*1/3*12&G (
6@W 4GMNC GODNMVFN ((=- K*8$%&G &- L%2*0V&6: 7*00&):< F- L%2*0V&6: &F$*02:< 7- L%2*0V&6: 1*4$&F$*02:h1*4$7*00&):< )- L%2*0V&6: 7:% *]20*4: $%8$%$2: & 6& $\B3$*0)&< *- L%2*0V&6: 7:% *]20*4: $%8$%$2: & 6& )*0*7X&< 8- 5&6:0 &F1:632:< ((+- K*43*120* B3*G
!a b " ! ,
ab
=
a b
((,- >*&% 6:1 $%2*0V&6:1 A [!5,12) A B [!9,5 ) < K*2*04$%* 6& 7&%2$)&) )* %;4*0:1 *%2*0:1 B3* /*02*%*7*% &6 7:%?3%2: A ! B < .*1/3*12&G O =
=
((J- >*&% 6:1 $%2*0V&6:1 %34I0$7:1 A "#!4 ,3 A B
)
=
B3* /*02*%*7*% &6 7:%?3%2: &- =
F- +
=
(!2 ,7"# < E& 7&%2$)&) )* %;4*0:1 !.LcU>
( B ! A) *1 $#3&6 &G 7- ,
)- J
*- M .*1/3*12&G F-
((M- K*2*04$%* *6 V&6:0 )* V*0)&) )* 6& 1$#3$*%2* /0:/:1$7$9%G ! x " !
x
2
=
x
.*1/3*12&G T K*17:4/:%#& *6 V&6:0 &F1:632: *% 6&1 1$#3$*%2*1 *]/0*1$:%*1G ((O-
3 x + 6
((Q- !2 x + 1 ((R-
x
2
! 5 x + 6
!"#$%& (O )* +,
(=T-
x
(=(-
x
2
! 4 x + 3
! x
x
(== x
(=+- P6 V&6:0 &F1:632: )* 6& )$8*0*%7$& )* ):1 %;4*0:1 *%2*0:1 /:1$2$V:1@ 1&F$*%): B3* 13 cYK *1 ,Q A B3* 13 134& *1 =QQ@ *1 $#3&6 &G &- RM F- ((= 7- (JT )- (R= *- =TT .*1/3*12&G )-
(=,- E& *]/0*1$9%
x
4
x
! x 2
2 =
!1
&- ! F- (
,0)
7-
(
, 1)
)-
(! #,!1) " (!1,0) {0}
! x *1 V"6$)& *% *6 1$#3$*%2* 13F7:%?3%2: )* %;4*0:1 0*&6*1G
"!
*-
!"
!
.*1/3*12&G 7(=J- >*&
A
=
{!1,0,1} A 6& :/*0&7$9% F$%&0$& *% A @ 2&6 B3* a * b
=
a +
b ! 1 G
`312$8$7&%): 13 0*1/3*12&@ )*2*04$%* 1$G &- 6& :/*0&7$9% *1 7:%432&2$V&< F- 6& :/*0&7$9% *1 &1:7$&2$V&< 7- 6& :/*0&7$9% 2$*%* *6*4*%2: %*320:< )- 6& :/*0&7$9% 2$*%* *6*4*%2: $%V*01:<
6@X !HVGHINEAD (=M- K*8$%&G &- L)*%2$)&)< F- P73&7$9%< 1"
(=O- >*& 6& *73&7$9% 6$%*&6
5"
x +
2 3
5 =
2 x @
&6 )*2*04$%&0 13 1:637$9% 7:%1$)*0&%): 7:4:
7:%?3%2: 0*8*0*%7$&6 & 6:1 %;4*0:1 0*&6*1@ 1* :F2$*%* B3* /*02*%*7* &6 $%2*0V&6:G &- [0,1) F- [1,2) 7- [2,3) )- [ 3,4 ) *- [ 4,5 ) .*1/3*12&G )(=Q- !&0& B3* 6& *73&7$9% 73&)0"2$7&G 4 x 2 ! 2kx + 1 = 0 @ 2*%#& >UEHYLo_ w_LYD *% *6 7:%?3%2: )* 6:1 %;4*0:1 0*&6*1@ 6& 134& )* 6:1 V&6:0*1 B3* )*F* 2*%*0 k @ *1 $#3&6 &G &- , F- + 7- = )- ( *- T .*1/3*12&G *-
!"#$%& (Q )* +,
(=R- >*&
Re
=
@ 0*13*6V& 6& *73&7$9% &1:7$&)& & 7&)& /0*)$7&):< E3*#: )* 7:4/0:F&0 131
!
()
()
1:637$:%*1@ 2&F36* 6:1 7:%?3%2:1 )* V*0)&) Ap x @ Aq x A n
( ) m !"a (m ! x ) q ( x ) : 5 x 6 1
&- p x F-
:
+
7-
( ):
r
x
2
bx#$ = b n % x
(
)
+
ax
=
32
x
+
( ) <
Ar x
+ 3 x + 2
!3=
x
!3
x + 1
()
.*1/3*12&G F- Aq x
(+T- >*& *6 7:%?3%2: 0*8*0*%7$&6
Re
=
!
A *6 /0*)$7&):
=
( )
" 7% #!1, ! & @ 7- Ar ( x ) 5' $ 2
p x : !4 x
=
{!4,2}
! 8 x !32 @ 6& 134& )* =
6:1 *6*4*%2:1 )*6 7:%?3%2: )* V*0)&) Ap ( x ) *1 $#3&6 &G &- Z=
F- T
7- =
)- Q
*- (= .*1/3*12&G &-
(+(- P6 V&6:0 )*
/&0& B3* 6& 134& )* 6&1 0&S7*1 )* 6& *73&7$9% 73&)0"2$7& 2 x + ( k ! 1) x + 9 = 0 1*& $#3&6 & Z+@ )*F* 1*0G
&-
k
1
F- Z,
3
7- Z+
)- +
*- , .*1/3*12&G *-
(+=- >*&
Re
=
!
A
()
p x : x
A 1* 734/6* B3*
1
2
! (m ! 1) x + ( m + 1) = 0 < >$
1 +
x
1
2 =
x
2
3
@ 7&6736*
1 3
m +
x
1
A
x
2
()
1:% 6:1 *6*4*%2:1 )* Ap x
< 3
.*1/3*12&G
(++- >*&
Re
=
!
A
()
p x :
&- T
x!
x + 29
F- (
=1
1 2
@ *%2:%7*1 N ( Ap ( x ) ) *1G
7- =
)- +
*- , .*1/3*12&G F-
(+,- >*& *6 7:%?3%2: 0*8*0*%7$&6
Re
=
A *6 /0*)$7&):
()
p u :
1 ! u +
u
=1
<
P%2:%7*1@ *1 5P.KDK B3*G &F7)*-
() ! Ap (u ) {1} N ( Ap ( u )) 1 N ( Ap (u )) 2 N ( Ap (u )) 3 Ap u
=
=
=
=
=
.*1/3*12&G )!"#$%& (R )* +,
(+J- >*& *6 7:%?3%2: 0*8*0*%7$&6
Re
A *6 /0*)$7&):
=
( )
p x :
4 x ! 3 !
x
!2
=
3 x ! 5 <
P%2:%7*1@ 6& 134& )* 6& 1:637$:%*1 )*6 7:%?3%2: )* V*0)&) Ap ( x ) *1 $#3&6 &G &- +
F-
1
7-
3
2
5
)-
3
*-
3
11 3
.*1/3*12&G &(+M- >*&% ! A
# :/*0&7$:%*1
F$%&0$&1 1:F0* *6 7:%?3%2:
S
=
Z @ )*8$%$)&1
/:0G
!a, b " S $%a # b = 3a + 2b + 1&' !a, b " S $%a # b = a 2 # ab + b2 &' P6 V&6:0 )* &- Z+
n
/&0& *6 73&6 1* 734/6* B3*G F- + 7- ,
4# n
=
2 ! n @ *1
$#3&6 &G
)- M
*- R .*1/3*12&G F-
(+O- >*& *6 7:%?3%2: 0*8*0*%7$&6
() q ( x) :
p x :
x x
2
+ x
2 =
Re
A 6:1 /0*)$7&):1G
=
! 56 = 0
49
P6 7:%?3%2: A"# p x ! q
( x )%$ *1 $#3&6 &G { ! 7, ! 8, 7} F- {!7, 7} 7- { ! 8,7} ()
&-
)-
{7}
{8}
*-
.*1/3*12&G )(+Q- >*& Re
=
! A
*6 /0*)$7&):
V*0)&) Ap( x) *1G &- Z, F- Z=
( )
p x : x
! x
7- T
=
x
2
6& >HcD )* 6:1 *6*4*%2:1 )*6 7:%?3%2: )* )- (
*- = .*1/3*12&G F-
(+R- >*& *6 7:%?3%2: 0*8*0*%7$&6
Re = !
+
@ 3% /:1$F6* V&6:0 )* k /&0& B3* 6& )$8*0*%7$& )* 6&1
1:637$:%*1 )* 6& *73&7$9% 73&)0"2$7& x 2 ! kx + 2 = 0 1*& $#3&6 & &-
F- !2
4
7-
0
)-
2
0 *1G
*-
2 2
.*1/3*12&G *(,T- >$ 6& 134& )* 20*1 %;4*0:1 /&0*1 7:%1*732$V:1 *1 =,@ *%2:%7*1 )*2*04$%* *6 /0:)372: )* 6:1 20*1 %;4*0:1< .*1/3*12&G ,QT (,(- K&)& 6& /0$4*0& *73&7$9%G x 2 + 5 x + 6 = 0 A 6& 1*#3%)& *73&7$9%G x 2 + 7 x + k = 0 < K*2*04$%* *6 /0:)372: )* 6&1 0&S7*1 )* &4F&1 *73&7$:%*1@ 1$ 1* 7:%:7* B3* 6& 0&S\ )* 4&A:0 V&6:0 )* 6& /0$4*0& *73&7$9%@ *1 2&4F$I% 1:637$9% )* 6& 1*#3%)& *73&7$9%< Y:%1$)*0* Re < &- Z=T F- Z(= 7- (= )- MT *- O= .*1/3*12&G )=
!"#$%& =T )* +,
(,=- K&):1 6:1 7:%?3%2:1 A =
{ x
! ! /
x
%;4*0: )* *6*4*%2:1 )*6 7:%?3%2: &- T
F- (
2
"
( A
7- =
}
3 x + 2 = 0
! B
A
{
B = x ! ! / x
2
"5 x +
} @ *6
6=0
) *1G )- +
*- , .*1/3*12&G )-
(,+- >$
Re
=
!@
()
&-
p x :
F-
q x :
()
)*2*04$%* *6 7:%?3%2: )* V*0)&) Ap x G 9
7
2 x + 6
()
x
2
!
5 x + 15
2 =
3
! 13 + x ! 13 = 0
()
.*1/3*12&G &- Ap x
(,,- H% %;4*0: &34*%2&): *% 131
5 24
=
! 33 $ " % @ F- Aq ( x ) # 20 &
=
{7}
*1 $#3&6 & 87 < K$7X: %;4*0: 1* *%73*%20& *% *6
$%2*0V&6:G &F-
!"70,71) !"71,72 )
)-
!"72,73) !"73,74)
*-
!"74,75)
7-
.*1/3*12&G 7(,J- P6 1"F&): #&12I 6&1 ):1 2*07*0&1 /&02*1 )*6 )$%*0: B3* 2*%S& *6 V$*0%*1< P6 ):4$%#: /0*12I 6& )I7$4& /&02* )* 6: B3* 4* 1:F09 *6 1"F&): A 4* B3*)&0:% f,J< E& 7&%2$)&) )* )$%*0: B3* 2*%S& *6 V$*0%*1 83*G &- f (=T F- f (JT 7- f =(T )- f =,T *- f +TT .*1/3*12&G F(,M- Y3&20: %$e:1 1* 0*/&02*% 7$*02: %;4*0: )* %&0&%?&1 )* 6& 1$#3$*%2* 4&%*0&G *6 /0$4*0: 0*7$F* 6& 4$2&) )*6 2:2&6 4*%:1 1*$1m *6 1*#3%):@ 3% 2*07$: )*6 0*12: 4*%:1 ):1m *6 2*07*0:@ 3% 73&02: )*6 %3*V: 0*12: 4*%:1 3%& A *6 73&02: %$e:@ 6&1 20*7* %&0&%?&1 0*12&%2*1< K*2*04$%* 6& 7&%2$)&) )* %&0&%?&1 B3* 6* 7:00*1/:%)*% & 7&)& 3%:< .*1/3*12&G +T (,O- H% #03/: )* &4$#:1 )*F*% 0*/&02$01* 3% )$%*0:< E& /0$4*0& V*\ B3* 6: $%2*%2&%@ 1* 0*/&02*% & f Q /:0 /*01:%& A 6*1 1:F0& f ,< P% 3% 1*#3%): $%2*%2:@ 1* 0*/&02*% & f R 7&)& 3%: /*0: &X:0& 6*1 8&62& f +< P%2:%7*1@ 6& 7&%2$)&) )* )$%*0: & 0*/&02$01* *1G &- f J= F- f MT 7- f MQ )- f OM *- f Q, .*1/3*12&G F!"#$%& =( )* +,
(,Q- >$ 4* 8&62& /&0& 2*%*0 f+,T *6 ):F6* )* 6: B3* 4* 8&62& /&0& 2*%*0 f=MQ@ 6& 7&%2$)&) )* )$%*0: B3* 2*%#: *1G &- f ,T F- f O= 7- f RM )- f (,, *- f (RM .*1/3*12&G *(,R- H% 20&F&?&):0 0*7$F$9 f +RT 7:4: /: /:0 *6 20&F&?: )* 3%& 1*4&%&@ 6&F:0&%): *% 2:2&6 ,Q X:0&1@ )* 6&1 73&6*1 ,T X:0&1 83*0:% %:04&6*1 A *6 0*12: X:0&1 *]20&1< P6 V&6:0 )* 7&)& X:0& *]20& *1 ($ &6 8$%&6 *6 *123)$&%2* :F23V: O+ /3%2:1@ )*2*04$%* 6& 7&%2$)&) )* /0:F6*4&1 B3* *6 *123)$&%2* 0*1:6V$9 F$*%< .*1/3*12&G +R (J(- K:1 ?3#&):0*1@ D A j< *4/$*\&% & ?3#&0< D6 7:4$*%\: )*6 ?3*#: D 2$*%* *6 ):F6* )* 6& 7&%2$)&) )* )$%*0: B3* 2$*%* j< E3*#: j 6* #&%9 ,TT )96&0*1 & D< >$ &X:0& j 2$*%* *6 ):F6* )* 6& 7&%2$)&) )* )$%*0: )* Dm *%2:%7*1@ D A j *4/*\&0:% & ?3#&0 0*1/*72$V&4*%2*@ 7:%G &- f QTT A f ,TT F- f JTT A f =JT 7- f MTT A f +TT )- f RTT A f ,JT *- f QTT A f (MTT .*1/3*12&G &(J=- P)3&0): 6* )$7* & c&0S&G x>$ &6 ):F6* )* 4$ *)&) 6* B3$2&1 *6 20$/6* )* 6& *)&) B3* 2*%S& X&7* ,T &e:1@ :F2*%)0"1 4$ *)&) &723&6xm 7&6736* 6& *)&) &723&6 )* P)3&0):< .*1/3*12&G MT &e:1< (J+- E& 134& *%20* 3% %;4*0: *%2*0: x A 13 0*7S/0:7: *1
26 5
@ 7&6736* *6 V&6:0 )*
5 x + 1 <
.*1/3*12&G =M (J,- E& 7$80& )* 6&1 )*7*%&1 )* 3% 7$*02: %;4*0: *]7*)* *% = & 6& )* 6&1 3%$)&)*1< >$ 1* )$V$)* *6 %;4*0: /:0 6&1 7$80&1 )* 131 3%$)&)*1@ 0*1362& 6:1
7 8
)*6 %;4*0: $%V*02$):< kY3"6 *1 *6
%;4*0:l .*1/3*12&G ,= (JJ- UF23V* 3% Q u )* &34*%2: 1* 13*6):@ 6: B3* 4* 1$#%$8$7: f=T,TT 4"1 &6 4*1< kY3"6 *0& 4$ 13*6): &%2*0$:0 A 73&%2: *1 4$ 13*6): &723&6l .*1/3*12&G f =JJ TTT@ f =OJ ,TT (JM- P% 3%& 6$B3$)&7$9% )* 6$F0:1 B3$*0: 7:4/0&0 (, 6$F0:1< D6#3%:1 73*12&% f +TT 7&)& 3%: A :20:1 f QTT 7&)& 3%:< kY3"%2:1 )* 7&)& 3%: /3*): 7:4/0&0 7:% f M =TTl .*1/3*12&G (T 6$F0:1 )* f +TT@ , 6$F0:1 )* f QTT< (JO- K*2*04$%* 3% %;4*0: *%2*0: 2&6 B3* 134&): &6 %34*0&):0 A &6 )*%:4$%&):0 )* +hQ 1* :F2*%#& MhJ 7:4: 0*1362&):< .*1/3*12&G Z++
!"#$%& == )* +,
(JQ- H%& 0*$%& A 3%& V$00*$%& )*)$7&% f O MTT 7&)& 3%& /&0& 1:7:00*0@ 7:% 6& 4$14& &A3)&@ & 3% 7$*02: %;4*0: )* %*7*1$2&):1< E& 0*$%& 1:7:00* & (JT %*7*1$2&):1 4"1 B3* 6& V$00*$%&@ /*0: I12& )& & 7&)& %*7*1$2&): f ($ &4F:1 &6F&e$6*1 20&F&?&% 1$4362"%*&4*%2*@ *%2:%7*1 *6 2$*4/: *% B3* 2&0)&0"% *% 7:%1203$0 6& /&0*) *1 $#3&6 &G &- 12
7
horas
F- 3
7
horas
7- 7
2
horas
)- 4
3
horas
*- 3
4
horas
.*1/3*12&G &(M(- H% $631$:%$12& /$)* B3* 3%& /*01:%& /$*%1* *% 3% %;4*0:< D6 0*1362&):@ 134&06* (Tm 4362$/6$7&0 /:0 =m 134&0 ,Mm )$V$)$0 /:0 =m 0*12&0 *6 %;4*0: B3* /*%19< >3 0*1362&): *1 ++@ kV*0)&)l kv3I 8:0436& 319l 2
.*1/3*12&G >S@
(
x + 10
2
)
+
46
! x
=
33
6@Y *EAHVGHINEAD (M=- K*8$%&G &- K*1$#3&6)&)< F- L%*73&7$9%< (M+- .*13*6V& 6&1 1$#3$*%2*1 $%*73&7$:%*1@ 7:%1$)*0&%): 7:4: 7:%?3%2: 0*8*0*%7$&6 6:1 %;4*0:1 0*&6*1< &-
3 2 ! x ! 15 " 0
F-
x
7-
x + 1
2
!9 "7 ! 1 " 2 x
(
.*1/3*12&G &- !3,7
)
C
& # @ F- !", !4$# % (! 2, 2 ) % &'4, +" @ 7- !"0,2#$ ' $
(
)
(M,- E& 7&%2$)&) )* %;4*0:1 *%2*0:1 B3* 1&2$18&7*% 6& $%*73&7$9%G &- =
F- +
7- ,
)- J
4 x ! 1
"
5 @ *1 $#3&6 &G
*- M .*1/3*12&G F-
!"#$%& =+ )* +,
(MJ- >*&
*6
7:%?3%2: !
( )
q x : x!
2
! <
! ! 3! $ & "4 4 %
4
0*8*0*%7$&6
=
!
A
6:1
/0*)$7&):1G
( )
p x : x ! !
"
0
A
c @ *%2:%7*1 *6 7:%?3%2: )* V*0)&) A p ( x ) ! Aq ( x ) *1 $#3&6 &G
! ! #4
F- " ,
&- # ,
Re
! ! "4
3! $ 4
% &
7- # ,
3! $ 4
&%
! ! "4
)- # ,
3! $ 4
*- !
& %
.*1/3*12&G &(MM- >*& Re
! A *6
=
/0*)$7&):
( )
p x : x " 2 ! 2 x @ *%2:%7*1
*6 7:%?3%2: )* V*0)&) Ap ( x ) *1
*6 $%2*0V&6:G &F7)*-
" 1% $ !2 3 ' # & " 2 % $! 3 2 ' # & ,
,
" 2 % $ 3 +!' & # # 2 & %$ !" 3 ( ' # 2 & %$ !" 3 ( ) *+ 2 +" ) ' ,
,
,
,
.*1/3*12&G 7(MO- >*& *6 7:%?3%2: 0*8*0*%7$&6
Re
A *6 /0*)$7&):
=
()
p x :
2 x x ! 4
"
8<
P6 7:%?3%2: )*
V*0)&) Ap ( x ) *1 $#3&6 &G
$ 16 ' ,3 , + !" # " & ( ) % 3 )( $16 ' F- (!",3) # & , +") %3 ( $16 ' 7- (!",4) # & , +") %3 ( &-
C
! 16 $ )- # 4, & " 3% ! 16 $ *- # 4, & " 3%
.*1/3*12&G 7(MQ- >*&
Re
=
!
A
()
p x :
x+
5 x + 3 x ! 1
0@
"
&-
( !",1) # ( 2, +")
F- [ !3, !1]
)-
(!3, !1) " (1, #)
*- 2, +!
+
(
*6 7:%?3%2: )* V*0)&) Ap ( x ) *1 $#3&6 &G 7-
[!3, !1] " (1, +#)
) .*1/3*12&G 7!"#$%& =, )* +,
(MR- >*& *6 7:%?3%2: 0*8*0*%7$&6
Re
=
!
x ! 3
( )
A *6 /0*)$7&):
p x :
2
x
!1
"
0 @
*6 7:%?3%2: )*
V*0)&) Ap ( x ) *1 $#3&6 &G
(!", !3] # (!1,1) # [3, ") (!", !2] # (!1, 0) (0,1) ! [ 2, ") (!", !3] # [3, ") (!", !2] # (!1,1) # ( 2, ")
&-
+
F7-
+
)-
+
*-
+
.*1/3*12&G &(OT- >*&
*6
( ) &- (
7:%?3%2:
q x : x!2
<
0*8*0*%7$&6
Re
=
!
A 6:1 /0*)$7&):1
!1 @ *%2:%7*1 *6 7:%?3%2: A ( p ( x ) F- (!4, !2)
, 4]
!" !
!
( )
p x :
x + 3 ! 1 "
( )) *1 $#3&6 &G )- (!2, 4)
A
0
q x
7- [!2, 4)
*- [!4, !2 ] .*1/3*12&G F-
(O(- >*& *6 7:%?3%2: 0*8*0*%7$&6
Re
=
!
A 6:1 /0*)$7&):1
()
p x :
x
2
!1 "
()
1
0 A q x : x
!1
<
1@
( ( ) ( )) *1 $#3&6 &G
*%2:%7*1 A p x
!q
&-
!1 2# " $
F-
(1 2) (!" 1) # (2 ") (!" 1#$ % (2 ") (!" 1#$ % &'2 ")
x
,
,
7)*-
+
,
,
,
,
+
,
,
+
.*1/3*12&G )(O=- >*& *6 7:%?3%2: 0*8*0*%7$&6 Re = !"0 , +#) A 6:1 /0*)$7&):1G
()
p x :
x
2
()
q x :
! 6 x " 0
x
! 1 ! 1
=
0
P%2:%7*1@ N " A p ( x ) ! q ( x ) $ *1 $#3&6 &G # %
(
&- T
)
F- (
7- =
)- +
*- , .*1/3*12&G 72
(O+- >*& *6 7:%?3%2: 0*8*0*%7$&6
Re
=
!
A *6 /0*)$7&):
( )
p x :
x ! 2 x 3
2
x ! 4 x + 4 x
"0
@ *6
7:%?3%2: )* V*0)&) Ap ( x ) *1G &F7-
(0, ) ( 2, ) (0,2 ) ( 2, ) +!
+!
!
+"
!"#$%& =J )* +,
)*-
{} ( ,2) !! 2 !"
.*1/3*12&G F(O,- >*& *6 7:%?3%2: 0*8*0*%7$&6
Re
()
" $ $ #
p x :
=
!
A *6 /0*)$7&):G
1 ! x
2 x + 6
+
x + 1
% ' es un número real ' &
()
P%2:%7*1@ Ap x *1 $#3&6 &G &-
"#!3 1
%$F-
(!3 1) (!3 !1) ! ("1 1) "#!3 !1) $ (!1 1%& (!1 1"#
7)*-
,
,
,
,
,
,
,
.*1/3*12&G )(OJ- H% &02*1&%: B3* 8&F0$7& A V*%)* 7&6\&): 2$*%* #&12:1 8$?:1 1*4&%&6*1 )* fMTT *%20* 1&6&0$:1 A :/*0&0$:1@ &6B3$6*0 )* 2&66*0 A 7:%134: )* *%*0#S& *6I720$7&< P6 7:12: *% 4&2*0$&6*1 /:0 7&)& /&0 )* \&/&2:1 *1 )* fQ@ 63*#: 6:1 V*%)* & 3% /0*7$: )* f(M< kY3"%2:1 /&0*1 )* \&/&2:1 )*F*% *6&F:0&01* A V*%)*01* 1*4&%&64*%2* /&0& :F2*%*0 32$6$)&)l &- c"1 )* JT /&0*1 )* \&/&2:1< F- c"1 )* OJ /&0*1 )* \&/&2:1< 7- c*%:1 )* OJ /&0*1 )* \&/&2:1< )- c*%:1 )* (TT /&0*1 )* \&/&2:1< *- P%20* JT A (TT /&0*1 )* \&/&2:1< .*1/3*12&G F6@87 *EKVHHIJE LGFALZFIHG (OM- P170$F& 6:1 &]$:4&1 )* !*&%:< (OO- P%3%7$* *6 2*:0*4& )* $%)377$9%< (OQ- P6 V&6:0 )*
! 59 ! #" 58 ! + 57 !
&- RT
33! +
32 !
+
F- R(
$ & *1G 31!%
7- R=
)- R+
*- R, .*1/3*12&G &-
(OR- K*43*120* B3*G
(QT- K*43*120* B3*G
!n " !,
!n ! !,
(
2 + 6 + 10 + … + 4n # 2
2
1
+
2
2
+
…+ n
2
(
)
n n+ =
=
2
2n <
)(
)<
1 2n + 1 6
!"#$%& =M )* +,
(Q(- H%& )* 6&1 :/7$:%*1 /0*1*%2&)&1 & 7:%2$%3&7$9% _U *1 3%& /0:/$*)&) V"6$)& /&0& 6:1 %;4*0:1 %&230&6*1<
(
)
n n +1
&-
1+ 2 + 3 +… + n
F-
( ) 1 + 3 + 5 + … + (2 n " 1) = 2 n " n ( n + 1) %
7)*-
=
2 + 4 + 6 + … + 2n
2 = n n +1
2
3
1
+
0 1!
2
3
+
1 +
2!
3
3
+
…+ n
2 +
+
3!
…+
3
=
n " 1 n!
$ #
=
2
' &
2 n! "1 n!
.*1/3*12&G 7-
(Q=- K*43*120* B3*G
(Q+- K*43*120* B3*G (Q,- K*43*120* B3*
!n " !, 1 4 7 … + (3n # 2) = +
()
p n :
+
( )( ) ( )( ) ( )( )
!n " !,
!n ! ! @
+
1 2
+
2 3
+
3 4
+
n
(3
n
)<
#1
2
… + ( n) ( n + 1) =
n
(
n+
)(
1
3
n+
2
)<
1* 734/6* 6& 1$#3$*%2* /0:/$*)&)G
1
1 +
1
1
+
(1) (3) (3) (5) (5) (7)
+
…
+
(2n ! 1) (2n 1) +
=
n
2n + 1
(QJ- P4/6*&%): $%)377$9% 4&2*4"2$7&@ )*43*120*G !n "!, n
(
n
2
+
)
2 *1 )$V$1$F6* /:0 +
6@88 ,[HEIHGD KA HNEFAN (QM- K*8$%&G &- C&72:0$&6< F- Y:4F$%&2:0$&< 7- !*0432&7$9%< )- Y:4F$%&7$9%< *- !0$%7$/$: &)$2$V:< 8- !0$%7$/$: 4362$/6$7&2$V:<
(QO- >*&%
3 2 ! x ! 15 " 0 Re
=
! A
p ( n ) :
( n 1)! ( n ! 2)! ( n ! 1) +
=
56 "
.*1/*72: & Ap ( n ) 1* /3*)* &8$04&0 B3*G &F7)*-
W$*%* 3% ;%$7: *6*4*%2:< W$*%* ):1 *6*4*%2:1< W$*%* 20*1 *6*4*%2:1< P1 V&7S:< P1 $#3&6 & Re < .*1/3*12&G &-
!"#$%& =O )* +,
(QQ- kY3"%2:1 *6*4*%2:1 X&A B3* 7:4F$%&0 )* ):1 *% ):1 /&0& B3* *6 %;4*0: )* 7:4F$%&7$:%*1 1*& (RTl .*1/3*12&G =T (QR- E& 7&%2$)&) n )* /*01:%&1 B3* )*F*% 7:4F$%&01* *% /&0*?&1 /&0& B3* *6 %;4*0: )* /&0*?&1 1*& (TJ@ *1 $#3&6 &G &- (, F- (J 7- (M )- (O *- (Q .*1/3*12&G F(RT- W0*1 &4$#:1 66*#&% & 3%& 7$3)&) *% 6& B3* 1:6&4*%2* X&A , X:2*6*1< >$ 7&)& 3%: B3$*0* X:1/*)&01* *% 3% X:2*6 )$8*0*%2*@ 6& 7&%2$)&) )* 4&%*0&1 *% 6& B3* /3*)*% 6:#0&06: *1G &- + F- , 7- (= )- =, *- ,Q .*1/3*12&G )(R(- >* )*1*&% 1*6*77$:%&0 + #&66$%&1 )* 3% #03/: )* J F6&%7&1 A , %*#0&1< E& 7&%2$)&) )* 4&%*0&1 *% B3* 1* /3*)*% 1*6*77$:%&0 + #&66$%&1 F6&%7&1 *1G &- M F- (T 7- (J )- =T *- MT .*1/3*12&G F(R=- !&0& 3% ?3*#: )* $%):08;2F:6 1* )$1/:%* )* 3% 2:2&6 )* (T ?3#&):0*1 N*%20* 2$236&0*1 A 13/6*%2*1-< >$ *6 *B3$/: 1* 7:%8:04& 7:% M ?3#&):0*1 2$236&0*1@ 6& 7&%2$)&) )* *B3$/:1 2$236&0*1 )$8*0*%2*1 B3* 1* /3*)* 7:%8:04&0 *1G &- MT F- (=T 7- =(T )- J T,T *- M TTT .*1/3*12&G 7(R+- P6 )$0*72:0 )* 6& 1*6*77$9% %&7$:%&6 )* 8;2F:6 )*F* 7:%8:04&0 *6 *B3$/:< >$ )$1/:%* )* + &0B3*0:1 A )*F* *17:#*0 (@ )* M )*8*%1:0*1 A )*F* *17:#*0 ,@ )* O V:6&%2*1 A )*F* *17:#*0 ,@ A@ 8$%&64*%2*@ )* , )*6&%2*0:1 A )*F* *17:#*0 =@ *%2:%7*1 *6 %;4*0: )* *B3$/:1 )$8*0*%2*1 B3* /3*)* 1*6*77$:%&0 *1G &- M JTT F- O =TT 7- Q ,TT )- R ,JT *- (T TTT .*1/3*12&G )(R,- H% #03/: )* &4$#:1 )*7$)* V$&?&0 A 0*&6$\&0 )*/:02*1 /6&A*0:1< P% 2:2&6 /:)0S&% V$1$2&0 O F&6%*&0$:1@ A /0&72$7&0 J )*/:02*1 )$8*0*%2*1< E& 7&%2$)&) )* 4&%*0&1 )$8*0*%2*1 *% B3* /:)0S&% *17:#*0 + F&6%*&0$:1 A 0*&6$\&0 = )*/:02*1 /6&A*0:1 *1G &- +J F- ,J 7- (TT )- +,T *- +JT .*1/3*12&G *(RJ- E& 7&%2$)&) )* 8:04&1 )$8*0*%2*1 B3* /:)0S&% 1*%2&01* *% 3%& 4$14& 8$6& )* &1$*%2:1@ = *73&2:0$&%:1@ + 7:6:4F$&%:1 A , V*%*\:6&%:1@ 1$ )*F*% 1*%2&01* ?3%2:1 B3$*%*1 2$*%*% 6& 4$14& %&7$:%&6$)&)@ *1G &- =, F- (,, 7- =QQ )- ( O=Q *- = OJM .*1/3*12&G )(RM- K:1 43?*0*1 A 20*1 X:4F0*1 )*F*% :73/&0 6:1 7$%7: /3*12:1 )* 3% &32:49V$6< >$ 3%& 43?*0 1* *%7&0#&0" )* 7:%)37$0 A 3% X:4F0* 1* 1*%2&0" *% *6 /3*12: )*6 7:/$6:2:@ 6& 7&%2$)&) )* 8:04&1 *% B3* 1* /3*)*% 3F$7&0 6:1 :73/&%2*1 )*6 V*XS736: *1G &- =, F- +M 7- ,Q )- O= *- (=T .*1/3*12&G F(RO- H% /$%2:0 2$*%* J 6&2&1 B3* 7:%2$*%*% /$%230&1 )* )$8*0*%2*1 7:6:0*1 *% 7&)& 3%&< E& 7&%2$)&) )* 4*\76&1 B3* /3*)* 0*&6$\&0 7:% *66&1 *1G &- (+ F- =J 7- =M )- =O *- (=J .*1/3*12&G 7!"#$%& =Q )* +,
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
& 20 # $$ ! ! 5
F-
& 20 # $$ ! ! 3
7-
& 17 # $$ ! ! 3
)-
& 17 # $$ ! ! 2
*-
& 15 # $$ ! ! 2 .*1/3*12&G )-
=TT- K* 3% #03/: )* O /*01:%&1 1* V& & *6*#$0 ( /0*1$)*%2*@ ( V$7*/0*1$)*%2*@ ( 1*70*2&0$: A = V:7&6*1@ I12:1 = ;62$4:1 7:% $#3&6 ?*0&0B3S&< E& 7&%2$)&) )* 8:04&1 *% B3* 1* /3*)* 66*V&0 & 7&F: *12& *6*77$9% *1G &- =( F- ,= 7- =(T )- ( =MT *- = J=T .*1/3*12&G )=T(- P% 3%& \&/&2*0S& 1* 2$*%*% J 4&*120:1 A Q :/*0&0$:1@ A 1* B3$*0* 20&F&?&0 *% #03/:1 7:%8:04&):1 /:0 J /*01:%&1@ /*0: 7:4: 4"]$4: )*F*% X&F*0 = 4&*120:1 *% *6 #03/:< E& 7&%2$)&) )* #03/:1 )$8*0*%2*1 B3* 1* /3*)*% 7:%8:04&0 *1 $#3&6 &G &- JM F- +JT 7- JMT )- R(T *- RMM .*1/3*12&G *=T=- H% *B3$/: /&02$7$/& *% (= /&02$):1 )* 832F:6 *% 3%& 2*4/:0&)&< P6 %;4*0: )* 4&%*0&1 *% 6&1 B3* *6 *B3$/: /3*)* 2*04$%&0 6& 2*4/:0&)& 7:% O V$72:0$&1@ + )*00:2&1 A = *4/&2*1 *1 $#3&6 &G &- O@R=T F- 12! 7- 7! ! 3! ! 2! )- OR= *- =@ROT .*1/3*12&G &=T+- >* 2$*%*% O ?3#&):0*1 )* 73&)0:@ M ?&0)$%*0:1@ J 6&%\&):0*1 A = 0*7*/2:0*1< Y&)& ?&0)$%*0: /3*)* :73/&0 73&6B3$*0& )* + /:1$7$:%*1 A 7&)& ?3#&):0 )* 73&)0: /3*)* :73/&0 73&6B3$*0& )* , /:1$7$:%*1< >$ 3% *B3$/: )* FI$1F:6 1* 7:%8:04& 7:% R ?3#&):0*1 A 7&)& 3%: )*F* :73/&0 6& /:1$7$9% B3* 6* 7:00*1/:%)*@ *6 %;4*0: )* 4&%*0&1 *% B3* /3*)* 1*6*77$:%&01* *6 *B3$/:@ *1G &- OT F- OTT 7- OOT )- OTTT *- OOTT .*1/3*12&G )=T,- H% #03/: )* &4$#:1 )*7$)*% 0*3%$01* *% 3% 0*12&30&%2*< >* *%73*%20&% $%)*7$1:1 *%20* O /6&2:1 83*02*1 A J 2$/:1 )* F*F$)&1< E& 7&%2$)&) )* 4&%*0&1 )$8*0*%2*1 *% B3* /:)0S&% *17:#*0 + )* *12:1 /6&2:1 83*02*1 A = )* *12&1 F*F$)&1@ *1 $#3&6 &G &- +J F- ,J 7- (TT )- +,T *- +JT .*1/3*12&G *=TJ- H% #03/: )* &4$#:1 )*7$)* V$&?&0 &6 U0$*%2* A 0*&6$\&0 )*/:02*1 *]20*4:1< P% 2:2&6 /:)0S&% V$1$2&0 M 1$2$:1 A /0&72$7&0 , )*/:02*1 *]20*4:1< E& 7&%2$)&) )* 4&%*0&1 )$8*0*%2*1 *% B3* /:)0S&% *17:#*0 , 1$2$:1 A 0*&6$\&0 + )*/:02*1 *]20*4:1@ *1 $#3&6 &G &- , F- (J 7- (R )- MT *- (=T .*1/3*12&G )!"#$%& =R )* +,
=TM- E& 7&%2$)&) )* 8:04&1 )$8*0*%2*1 *% B3* 1* /3*)*% 1*6*77$:%&0 , 4:%*)&1 )* 3% 2:2&6 )* M *1 $#3&6 &G &- , F- (T 7- (J )- =, *- +MT .*1/3*12&G 7=TO- P% 3%& *6*77$9% )* 3% 7:4$2I 1* *17:#*% )* *%20* (T /*01:%&1@ J /*01:%&1 B3* 6: 7:%8:04*% A )* *1* 7:4$2I 1* *6$#*% & + /*01:%&1 /&0& 3%& )$0*72$V&< E& 7&%2$)&) )* 4&%*0&1 *% B3* 1* /3*)* 0*&6$\&0 *12& *6*77$9% *1G &-
10!
10!
F-
4!
10!
7-
6!
10!
)-
( 2) 6!
*-
( 2)5!
10!
( 2) 4! .*1/3*12&G )-
6@86 ,ANCALG KAM OIENLIN =TQ- P%3%7$* *6 2*:0*4& )*6 F$%:4$:< =TR- kY3"6 *1 *6 )*1&00:66: )*
(a ! b)
5
l
=(T- g&66* *6 2I04$%: 73&02: )*6 )*1&00:66: )*
( x
+
2 y
)
5
12
! 1$ =((- K*2*04$%* *6 2I04$%: 7*%20&6 )* # x + & " x %
=(=- DV*0$#y* B3I V&6:0 )*F* )&01* & 6& V&0$&F6* x /&0& B3* *6 2*07*0 2I04$%: )*6 )*1&00:66: )* 5
"3 % $ ! x ' 1*& $#3&6 & RT< # x &
" =(+- P% *6 )*1&00:66: )*6 F$%:4$:G $ t ! # 2
&- Z=J=
F- Z(O=
10
% ' @ *6 7:*8$7$*%2* )*6 2I04$%: 7*%20&6 *1 $#3&6 &G t &
1
7- Z(=M
)- (O=
*- =J= .*1/3*12&G &-
=(,- E& >HcD )* 6:1 *]/:%*%2*1 )* F$%:4$:
(
12
a
+
3
2 b
)
*1
&- 5*0)&)*0:
35 6
a
A b /&0& *6 1*#3%): 2I04$%: *% *6 )*1&00:66: )*6
< F- C&61: .*1/3*12&G &-
=(J- >$ *6 7:*8$7$*%2* )* V&6:0 )* &- (
a
x
& *% *6 )*1&00:66: )*6 F$%:4$: $ %
7
1
x + ax
2
# ! "
*1 $#3&6 &
7 3
@ *%2:%7*1 *6
@ *1 $#3&6 &G F- =
7- +
)- ,
*- R .*1/3*12&G 7-
!"#$%& +T )* +,
=(M- P6 2I04$%: B3* %: 7:%2$*%*
&-
F- !56 yz
56 yz
! 1 3 *% *6 )*1&00:66: )*6 F$%:4$: # ! # 5 xyz "
x
7-
56
)- !
yz
56
*- !
yz
$8 x & *1G & %
56 5
yz
.*1/3*12&G )-
# x =(O- >$ *6 73&02: 2I04$%: )* % " $3
& ( 7:%2$*%* x '
3
n
2
x
2
< L)*%2$8$B3* 6& /0:/:1$7$9% 5P.KDKP.D<
&- n " (0,6) F- P% *6 )*1&00:66: )*6 F$%:4$: *]$12*% M 2I04$%:1< 7- P6 73&02: 2I04$%: *1 *6 2I04$%: 7*%20&6 )*6 )*1&00:66: )*6 F$%:4$:< )- P6 )*1&00:66: )*6 F$%:4$: 2$*%* ):1 2I04$%:1 7*%20&6*1< *-
n
2
<
3
" =(Q- >$ 1* 2$*%* 6& *]/0*1$9% $ m ! #
5
4 m
2
% ' @ *%2:%7*1 &6 )*1&00:66&0 *6 F$%:4$:@ *6 7:*8$7$*%2* )*6 &
2I04$%: B3* 2$*%* m!4 *1 $#3&6 &G &- ZM,T F- Z+=T 7- =JM
)- M,T
*- (@=QT .*1/3*12&G &-
! =(R- K*2*04$%*@ 1$ *1 /:1$F6*@ *6 7:*8$7$*%2* )*6 2I04$%: )*6 F$%:4$: # a "
5
2
b$ +
& B3* 7:%2$*%* ! < a% "
.*1/3*12&G _: *1 /:1$F6*< 6@8? "VHADINEAD ==T- K*8$%&G &- >37*1$9%< F- !0:#0*1$9% &0$24I2$7&< 7- !0:#0*1$9% #*:4I20$7&< ==(- E& 134& )* 6:1 *%2:%7*1 % *1G &- (T
/0$4*0:1 2I04$%:1 )* 6& /0:#0*1$9% &0$24I2$7&
n
F- (=
7- ((
)- R
9 12 15 ,
,
,...
*1
306 @
*- (O .*1/3*12&G F-
===- P% 6& 1$#3$*%2* 137*1$9%G 671, 665, 659, … @ )*2*04$%* 73"6 *1 *6 2*07*0 %;4*0: %*#&2$V:< .*1/3*12&G Z(+
{
}
==+- >*& 6& /0:#0*1$9% &0$24I2$7& 6,19,32,… < E& 7&%2$)&) )* 2I04$%:1 B3* *12"% *%20* =TT A +JT *1 $#3&6 &G &- (T
F- ((
7- (=
)- (+
*- (, .*1/3*12&G 7!"#$%& +( )* +,
==,- >$ 6:1 /0$4*0:1 )$*\ 2I04$%:1 )* 3%& /0:#0*1$9% &0$24I2$7& 134&% +J A *6 /0$4*0 2I04$%: *1 (T@ *6 )I7$4: 2I04$%: *1 $#3&6 &G &- J F- + 7- Z+ )- , *- ZJ .*1/3*12&G 7==J- K* 6:1 1$#3$*%2*1 2I04$%:1 )* 3%& /0:#0*1$9% &0$24I2$7&G 2I04$%:1 B3* *12"% *%20* =TT A JTT *1 $#3&6 &G &- == F- =+ 7- =,
)- =J
{7 20 33 } @ 6& 7&%2$)&) )* ,
,
,...
*- =M .*1/3*12&G F-
==M- E:1 /:1 4*%13&6*1 )*6 /0I12&4: B3* z:6&%)& X$\: &6 F&%7: 8:04&% 3%& /0:#0*1$9% &0$24I2$7&< >$ *6 1*#3%): /: *1 )* f (=O A *6 B3$%2: /: *1 )* f (J,@ *%2:%7*1 *6 )I7$4: /: *1G &- f (OT F- f (O= 7- f (R, )- f (RR *- f =TQ .*1/3*12&G )==O- E&30& X& 1$): 6& #&%&):0& )* J /0*4$:1 )* )$%*0: *% *8*72$V:< E& 134& 2:2&6 )* 6:1 /0*4$:1 83* fJTTT< >$ X3F: 3%& )$14$%37$9% )* f(TT *%20* /0*4$:1 137*1$V:1@ *%2:%7*1 *6 /0*4$: )* 4&A:0 V&6:0 83*G &- f ( TTT F- f = TTT 7- f = QTT )- f + TTT *- f ( =TT .*1/3*12&G *==Q- >$ E3$1 7:4/0& 0*#&6:1@ ):%)* *6 /0$4*0: 6* 7:129 f (T@ f (, *6 1*#3%):@ f (Q *6 2*07*0:@ A *6 7:12: )* 7&)& 0*#&6: *1 f , 4"1 B3* *6 &%2*0$:0@ *%2:%7*1 E3$1 / /:0 6:1 JT 0*#&6:1G &- f J (TT F- f J =TT 7- f J +TT )- f J ,TT *- f J JTT .*1/3*12&G )==R- H% X:4F0* &73*0)& /&0 3%& )*3)& )* f+MTT *% ,T /:1 1*4&%&6*1 B3* 8:04&% 3%& /0:#0*1$9% &0$24I2$7&< Y3&%): +T )* 6:1 /:1 *12&F&% 73F$*02:1@ *6 $%)$V$)3: 8&66*7* )*?&%): 3%& 2*07*0& /&02* )* 6& )*3)& 1$% /&0< Y&6736* *6 V&6:0 )*6 /0$4*0 /:< .*1/3*12&G f(+Q =+T- >* )*1*& 7:%1203$0 3%& *17&6*0& 7:% F6:B3*1 )* /$*)0& 2&6 7:4: 1* 43*120& *% 6& 8$#30& &)?3%2&< P6 %;4*0: )* F6:B3*1 B3* 1* %*7*1$2&% /&0& B3* 6& *17&6*0& 2*%#& =Q *17&6:%*1@ *1 $#3&6 &G &- (+T F- OTT 7- OQ, )- (M=, *- +=,Q .*1/3*12&G )=+(- >$ Y&06:1 7:4/09 JT 6$F0:1@ ):%)* *6 /0*7$: /:0 6$F0: 83*G fQ *6 /0$4*0 6$F0:@ f(( *6 1*#3%): 6$F0:@ f(, *6 2*07*0 6$F0:m A )* *12& 4&%*0& *6 7:12: )* 7&)& 6$F0: 83* f+ 4"1 B3* *6 /0*7$: )*6 6$F0: &%2*0$:0@ 7&6736* 6& 7&%2$)&) )* )$%*0: B3* Y&06:1 / /:0 6:1 JT 6$F0:1< .*1/3*12&G f,@TOJ =+=- E& 134& )* 6:1 1*$1 /0$4*0:1 2I04$%:1 )* 3%& /0:#0*1$9% #*:4I20$7& *1 $#3&6 & R V*7*1 6& 134& )* 6:1 20*1 /0$4*0:1 2I04$%:1< Y&6736* *6 V&6:0 )* 6& 0&\9%< .*1/3*12&G = !"#$%& += )* +,
=++- H% *]&4*% 2$*%* (TT /0*#3%2&1< >$ 6& /0$4*0& /0*#3%2& 2$*%* 3% V&6:0 )* = /3%2:1 A 7&)& 3%& )* 6&1 1$#3$*%2*1 V&6* , /3%2:1 4"1 B3* 6& &%2*0$:0@ *12* *]&4*% 2$*%* 3% V&6:0 )*G &- =TTT /3%2:1 F- =T=T /3%2:1 7- ,TTT /3%2:1 )- =TTTT /3%2:1 *- =T=TT /3%2:1 .*1/3*12&G )=+,- >*& 6& 137*1$9% 3,6,9,12,15,… E& 134& )* 6:1 (TT /0$4*0:1 2I04$%:1 )* *12& 137*1$9% *1 $#3&6 &G &- (J@(+Q F- (J@(,( 7- (J@(,, )- (J@(,O *- (J@(JT .*1/3*12&G *=+J- E& 134& $%8$%$2& )* 6:1 2I04$%:1 )* 3%& /0:#0*1$9% #*:4I20$7& )*70*7$*%2* *1 $#3&6 &6 73")03/6: )* 13 /0$4*0 2I04$%:< !:0 6: 2&%2:@ 6& 0&\9% )* 6& /0:#0*1$9% 2$*%* 3% V&6:0 )*G &-
1
F-
2
1
7-
4
3
)-
4
1
*-
8
7 8
.*1/3*12&G 7=+M- E& 134& )* 6:1 /0$4*0:1 1$*2* 2I04$%:1 )* 3%& /0:#0*1$9% #*:4I20$7& )* 0&\9% + *1 OMJ(< K*2*04$%* *6 /0$4*0: A *6 1I/2$4: 2I04$%:< .*1/3*12&G !0$4*0 2I04$%:G O@ >I/2$4: 2I04$%:G J(T+ =+O- E& 134& )* 3%& /0:#0*1$9% #*:4I20$7& )*70*7$*%2* $%8$%$2& 2$*%* 3% V&6:0 &/0:]$4&): )* 2 <
>$ 13 /0$4*0 2I04$%: *1 $#3&6 &
&- !
2 2
F- !
1
7-
2
2 2
@ *%2:%7*1 6& 0&\9% )* )$7X& /0:#0*1$9% 1*0"G
1
2
)-
2
*-
2
2!
2
2
.*1/3*12&G 7=+Q- Y:%1$)*0* 6& 134& $%8$%$2&G S = x + x 2 + x 3 + x 4 + ... >$ S *1 &/0:]$4&)&4*%2* $#3&6 &G x
&-
x
!1
F-
1
7-
x + 1
x
)-
1 ! x
0<
x < 1
x
(1 ! )
2
@ *%2:%7*1 *6 0*1362&): )* *-
x
1 ! x 1 + x
.*1/3*12&G 7=+R- P6 V&6:0 )* 6& *]/0*1$9%G 1 ! &-
3 4
F-
4 3
1 2
2
1 +
2
! 4 7-
1 2 4 5
6
1 +
2
! 8
1 10
2
± ...
)-
5 4
@ *1 &/0:]$4&)&4*%2* $#3&6 &G *-
9 8
.*1/3*12&G 7-
!"#$%& ++ )* +,