!#"%$'&)(+*,-*/.103254/6798:0<;>=638?0A@CB7D=3EF$
G
$%d 4 H3I 4 JLKNMPO/QSRTUO/JVO/WO3QSR3OXNY 16 JZKUMPO3QSRMPWU[]\SJVO/WP^_O`PR3Qa[DbR1\SR3QXNY 16
MPWU[c\aJ
TUO1\e[fJ1gP\+TRO/QSRO
$ G O/WP^_`PR3Qa[DbR1\SR3QeO3QSRWUMPbRQS[DhiOjfj IR/KNM[]kNOjlRWN\#mf[ln1WPXUQa[DWPn
MPWU[c\aJXNY?bRO/J+MPQSR3bR3WN\po
QSRV\+TR3QSR%O/WIX\+TR3QqQSR3hS\SO3WPnAjlRJsr![]\+T[DWN\SR/n1QaOjt^N[DbR3WPu
Jv[DXUWJ-\+TUO\-$ JZTUO/QSR'\+TN[DJF`QSXU`)RQa\wIPxy9Y`)XUJvJv[DHUjDRUg3JZTX/rz\+TUO1\?IUXNM{TUO1kUReY9XNMPWP^<OjfjJvM)hvT
Rv|3O3b`UjlRJ
BU*3}p~7Dw*46_1*/.?>
0 ~f.
# =AAZ:=3(i(v0A(v*/tUw301(p.v7l=U7{&)*3}cp730 SUwa=2P0=U7l=1}a~3)6=
:=/(++0L}*=3BA,?=D$
y9YtO%QSR/hS\aO/WPnAjDR#r![c\+TJv[D^PRJ x O/WP^ y TUO/Jq\+TR<nA[]kURW`QSXU`)RQa\wI?g\+TR3W{r<R!bMPJi\
TUO1kUR 2(x + y) = xy /\+TUO1\[DJAg
$
2x
4
y =
d
= 2+
x−2
x−2
TRQSRY9XUQSRNg?$\+TRXUWUj I_`)XUJ+[c\S[ kUR{[fWN\iR3nR3QaJ x Y9XUQsr<TN[DhvT y [DJ#O`)XUJ+[c\S[ kUR{[fWN\iR3nR3QeO/QSR
3 g 4 g)O/WP^ 6 ly \'$ Y9XNjcjlX/r%J'\+TUO1\O 3 × 6 QSR/hS\aO/WPnAjDR[fJ\+TR{XUWUj I5X\+TR3QXUWPR!r![]\+T\+TR
nA[]kURW`QSXU`)RQa\wI
U aai]++ ¡v¢?U £A¤i¥L¦1§i¨D¤L -©v©?aªl«Z a¥ª ¦3¬v®#+ª ¯L +° aª ¦±L¤Sca¥1¡v¤«S§i¦² ¥ +¥1a9° ¤a³N´L¡a¨ª
µ ° ° ¥Lw¶v° ¦·)«Zª]¨wa¸+¥1ª ¦¹)ºv³¤i¥ s»N«iaª ¤Z+¼3¨D° ¸q¶a¨D¦®3¥Z°cia¨D9°c v¤Z sº+)½1º1½+¦ µ «Za¥+NN° ¨wa9¦LU¨f§La¥1ª ° ¥¤¾
!!¿V$'&)(+*,-*/.103254/6798:0<;>=638?0A@CB7D=3EF$
$ G
SQ [DnT\+uvO3WPnAjlR3^\iQa[DO3WPnAjlR{TUO/JjDR/n1J#XNYjlRWPn\+T a O3WP^ b $hiÀ [fQShijlR{XNY-QaO^N[fMPJ r
\iXNM)hvTRJs\+TRV\pr<XjlR3n1J'O/WP^TUO3Js[c\aJh+RWN\iQSR<XUW{\+TR<T/IU`)X\SR3WUMPJZR
TX/r\+TUO1\
G
1
1
1 $
+
=
a
b
r
BU*3}p~7Dw*546Á~.7D=U3*Ât(al@ÃU01(SsN30A(a.+2N=N2/ÄÅ)?Å~U0A?*/.FÄD(v0.ZUÄs(pÆU0At7DDP=$
ywW 4ABC gjlRA\ a O/WP^ b ^R3WP$È
X\SÇ R\+TR
Jv[D^PRJ BC O/WP^ AC gsQSR3Jv`)R3hS\S[]kUR1j]I
R1\ T
HPRV\+TR%`PXN${[DWNÀ \r<TR3QSRV\+TR<hi[fQShijlR#\iXNM)hvTRJs\+TR
Jv[D^PR AC [fWPh+R 4ABC O/WP^ $-d 4AOT O3QSR
Jv[fb[cjDO3QLg3r<R%TUO1kUR a = r
TRW
b
ab − ar
ab
1
r
=
B
r
b+a
1
1
=
+
ab
a
b
O
r
b−r
= rb
= r(b + a)
....................................
...........
........
........
......
.....
....
.....
....
.
.
...
...
.
...
...............
..
..
....... ..................
..
..........
.....
..
..........
..
..
.
...............................................
...
..
.... ..................
......
..
..........
.
........
...
.
.......... .
..........
..
.... ...
.
..
.... ....
... ...................
.
..........
...
... ....
....
..........
...
....
.... .......
..........
.....
.
.......
..........
.
...
.
.
.
..
..........
.........
....
.
.
.
.
....
.
.
.
.
.
.
.
.
.
.........................................................................................................................................................................................................
C
T
$
U a-aici+ q¡v¢:U £A¤i¥L¦1§i¨D¤L -©+©taªD«L a¥1ª ¦¬Z®+ª]¯1L +° aª ¦/±L¤i]a¥¡v¤L«i§S¦²]¥1 +¥a9° ¤A¾
A
1
!%$'&)(+*,-*/.103254/6798:0<;>=638?0A@30A=/@$
[ kUR3W O hiM)HPRzr![c\+T [c\aJ R1[lnT\
h+XUQSWPR3QaJ hiMU\>X H3I `UjDO3WPR3JAg5TX/r bO3WI
^U[fOnXUWOjDJwXN[DWU[fWPn\+TR 24 WPRAr
h+XUQSWPR3QaJjf[DR%h+XUb`jDR1\SRj I[fWJv[D^PR#\+TR<hiM)HR/x
BU*3}p~7Dw*4/6Á0A?0/30 30{3=}w*?2P0;>=3.v.10A6$
G
.........................
....... ............ ..............
........
........
.
........
........
...
.......
........
.
.
.
..
.
.
.
......
...
..
......
......
.
.. ..
.......
.
.
. ..........
.
.
..... ...
.
.
. .
........ ..............
.
...
.
.
.......... ...............
.
.
.
.
....
.
...
...
........
.
.
.
.
.
.
.
........
...
.....
.....
...............................................
...
...
...... . ...
... ...
...
...
.
...
...
......................
.
.
....
.....
....
.
.
.....
.....
.
.
....
..
..
...
.
.
.
.
.
....
.....
..
...... ......
.
.
.
.
.
.
.
.
..... ...
.
... ......
.
.
.
.
.
... .
.....
.....
.....
........
....
....
........
........
....
........
........
........
........
........ .................. ..............
.
.
.
..................................
^U[fOnXUWOjr![cjfj3jf[DRs[DWJ+[l^Rs\+TRWPR
r )nAMPQS$ R'[fY[c\wXN[fWJ
\pr<X<kUR Qa\S[Dh+RJs\+TUO1\qO/QSR!WPX\sXUW{\+TR%JvO/bRVYlOhiR
O/hvT'kUR3Qp\iR+|'[fJXUW 3 YlOhiR3J O\iQS[fO/WPnAMjfO/Q)YlO/h+Rr<TR3QSR
\+TR{hiXUQiWPRQ'XNY:\+TRhiM)HPR%MPJLR3^\iXHPRNgPO3WP^\pr<XXhS\aOnXUWOj $
dYlOhiR3JFr<TN[DhvT{O/QSRVQSR3bWO/WN\aJXNY\+TR%XUQa[lnA[fWO3jPJZKUMPO3QSRVYlO/h+RJ
TNMPJAgR3O/hvTkURQa\SRv|s[DJXUWe\+TRsJvO/bRqYlO/h+RqO/J 7 kURQa\S[Dh+RJtYlQSXUb
XUWPRXNYs\+TRXhS\aOnXUWOjqYlO/h+RJ{O/WP^ 6 O/^P^N[c\S[DXUWO3jFkURQa\S[Dh+RJ
YlQSXUb\+TReX\+TR3QXhS\aOnXUWOj/YlO/h+RemDJv[fWPh+Rs$ \+TdRs\pr<X!XhS\aOnXUWOj
YlOhiR3J5JLTUO3QSR>O>Jv[D^PR
O3WP^ 2 kURQa\S[Dh+RJio
TRkURQa\S[Dh+RJYlQSXUb \+TR\iQS[fO/WPnAMjfO/QYlO/h+R $
dTUO1kURVOjDQSRO^/IHR/RWh+XNMPWN\SR/^{O3bXUWPn\+TRekURQa\S[Dh+RJYlQSXUb \+TRe\pr<X{XhS\aOnXUWOjUYlOhiR3J
TRQSRY9XUQSRNgR3O/hv$ T'kUR3Qp\iR+|'[fJh+XUWWPR3hS\iR3^VH3I[fWN\iRQiWOj/^N[DO/nXUWO3jfJ:\SX 24−7−6−1 = 10
X\+TR3Q- kUR3Qp\S[lhiR3J
RWPh+RNg!\+TR
$ \iX\aO3jVWUMPbHR3Q_XNY%[fWN\iRQiWOj<^N[DO/nXUWO3jfJXNY!\+TRWPRA
r :nAMPQSR[fJ
1
(24
2
· 10) = 120
!%$'&)(+*,-*/.103254/6798:0<;>=638?0A@CB7D=3EF$
y9YtOjfj)`Ujf[f
W J'O/QSR!`jDXU
W J'O/WP^JLXUbR!`jcMPWJO3QSR%`Ujf[fWJAg/r<TN[lhvTXNY)\+TR%Ji\SO1\iR3u
bRWN\SJggVbMPJ+\sHRV\iQaM)R/$ x
% jc[DW/J % jDXUWJ
G À jfjP`jc[DW /JO3QSR%`UjfMPW /J $
....
....
................. ............................................. ........................
........
..
.....
.....
..... .........
....
....
....
....
....
! À XUbR!`UjlXUW /JO3QSR%`UjfMPW /J $
.
.
.
.
...
..
.. 4 ...
..
..
..
..
..
1
2
...
...
..
..
..
..
.
.
.
..
XUbR!`Ujf[fW J'O/QSR!WPX\q`jcMPW J
..................................................
....
..
.
.......
......
..
.
.
.
.
...
.
...
. .......
.... ...
.
..
.
.
.
....
..
..
.
..
BU*3}p~7Dw*4/6Á0A?0/30 30{3=}w*?2P0;>=3.v.10A6
$
.
.
...
...
.
.
.
.
.
...
7 .. .. ..
.
..
" XUWJ+[l^R3Q\+TR bXUJi\nRWPR3QaOjr%OAI\+TUO\\+TR3JZR
\+TUQSR/R
$$JL# RA\SJhiO3WHRQSRjfO\SR/^>[fW>\+TR^N[DO/n1QaO/b XUW
\+TR
...
...
..
.
.
.. .....
.... ...
....
...
.. ....
..... ..
..... .......
.. ...
.........
...........
.
.
.
.........
.
.
.
.................
........................... ....................................... ...
...
.
..
.
.
..
...
...
...
...
....
....
.....
...
.
.
........
.
.
.....
............
............................
6
5
3
% jcMPWJ
QS[DnT\
R%r![cjfj?MPJLR n(k)$ \iXQSR3`QSR3JZR3WN\'\+TRWUMPbHPRQ
XNY?bRÀ bHR3QaJs[DWQSR3nA[lXUW k
$sÀ
[DWPhiR<Ojfj:`Ujf[f
W JO/QSR%`jDXUWJAgr<R<TUO1kUR n(1) =$ n(6)
= 0 [fb[cjDO3QSj I?gUJv[fWPh+R
%
JLXUbR`jcMP
W JVO/QSR`Ujf[f
W JAg:\+TRW n(6) +
$ ' n(7) 6= 0 MU\a\S[DWPn<\+TN[fJ\SXnR1\+TRQ'r![]\+T
\+T&
R )QaJi$ \qhiXUWP^U[]\S[lXUW?g3r<R!nRA\ n(7) 6= 0 X/r r<R#jDX(
X %O\R3O/hvTXNYP\+TR!Ji\SO1\iRbRWN\SJ
eg g À
\SO1\iRbRWN)
\ >[DJeR3KUM[ kNO3jDR3WN\\SX n(1) = n(4) =$ 0 gNr<TN[DhvTbOAIXUQqbO1IWPX\
HPRV\iQaM)À R%J+[DWPhiR#r<R%TUOAkUR%WPX{[DWUY9XUQSbO1\S[lXUWXUWQSR/nA[DXUW 4
$ À
R WPX/r \+TUO1\
\SO1\iRbRWN
\ [DJR3KUM[ kNO3jDR3WN\#$ \SX n(5) + n(7) 6= 0 [fWPh+Rr<*
n(7) 6=
À 0 gN\+TN[fJ'J+\aO\SR3bR3WN\[fJq\iQSM)R
,$ #
\SO1\iRbRWN+
\ [fJqR/KNM[]kNOjlRWN\\SX n(1) +$Fn(4)
6= 0
R /WPX/r \+TUO1\ n(1) = 0 g
d
HUMU\Fr<R<$ TUOAkUR<WPX{[DWUY9XUQSbO1\S[lXUWXUWQSR3nA[lXUW 4 TN[fJebR3O3WJs\+TUO
\ !bOAIXUQqbOAIWPX\
HPRV\iQaM)d R
$
TNMPJAgN\+TR%XUWUj]IJi\SO1\iRbRWN\F\+TUO\qbMPJ+\sHRV\iQSM)RV[DJ
U aai]++ V¡v¢sU £A¤i¥L¦:§i¨D¤L e©v©saªl«Z a¥ª ¦t¬Z®+ª]¯1L +° aª ¦-±L¤Sca¥1¡v¤«S§i¦-²]¥1 +¥a9° ¤S³¤i¥ ´L¡a¨ª µ ° ° ¥w¶+° ¦·P«Lª ¨wa¸+¥ª ¦N¹Pº3¾
#!%$'&)(+*,-*/.103254/6798:0<;>=638?0A@CB7D=3EF$
[ kUR3W!O3WIVJ+\aO/Qp\S[DWPnqWUMPbHR3QmlX\+TR3Q)\+TUO/W 1o+g3nRA\tOWPRrzWUMPbHR3QtH3I%^N[]kN[l^N[DWPn
\+TRWUMPbHPRQ 1 jDO3QSnR3Q%\+TUO/W
$'IUd XNMPQ{Ji\SO3Qa\S[fWPnWUMPbHR3QH3I
\+TRWUMPbHPRQ 1 JZ$b# O3jcjlRQ
\+TUO/WIUXNMPQs Ji\SO3Qa\S[fWPn!WUMPbHR3Q
TR3W^PX{\+TR<JZO3bR#r![c\+T{\+TN[fJeWPRr WUMPbHPRQ
TUO1\
TUO/``)RWJLx |`UjDO[D
W BU*3}p~7Dw*4/6Át=U4(a0L}Ât(aD@Ã0A(S:Æ/(a=U2P
0 e.17S~U2P0At7l~U0A?*/.Äl(+0A.vÄs(aÆN01?7Dl=P$
#
$
R XH\SO[DW{\+TR!J+\aO/Qp\S[DWPnVWUMPbHPRQ
<
$
ywWP^PR3R/^:g?jlRA\ x 6= 1 HR\+TRJ+\aO/Qp\S[DWPnWUMPbHPRQ ywW5\+TR{JLR3h+XUWP^_Ji\iR`r<RnRA\
\+TR%WUMPbHR3Q
x+1
2x
+1
2x
x−1
x−1
=
=
= x
x+1
2
2
−1
x−1
x−1
g
$
O/JhijDO[DbR/^
U aai]++ ¡v¢?U £A¤i¥L¦1§i¨D¤L -©v©?aªl«Z a¥ª ¦3¬v®#+ª ¯L +° aª ¦±L¤Sca¥1¡v¤«S§i¦² ¥ +¥1a9° ¤a³N´L¡a¨ª
i¤«i¨D¤¬Lªcp° ¦aªD«L a¥1ª ¦¬+¤i
¸ f+¨w #®¥L°]+a¨D9°]ª]¢S¦ µ ° ¨l¸° ¥L§i¯¤i¸¦U ¤L¡v¤i¸q¤i¦
µ ° ° ¥Lw¶v° ¦N·P«Lª ¨wa¸+¥ª ¦N¹)ºv³¤i¥
®/¬pt¾
##$'&)(+*,-*/.103254/6798:0<;>=638?0A@CB7D=3EF$
G
h+QSXUJZJvWUMPbHR3Q-[fJjc[ /RO!$,h+#QSXUJvJir<XUQS^Rv|/hiR3`N\-\+TUO1\\+TR#O/WJSr<R3QaJO/QSRWUMPbHR3QaJ
r![c\+T%XUWPR#^N[lnA[]\[fWROhvT!JLKNMPO/QSR
TUO\t[DJ-\+TReJ+MPb XNYUO3jcj/\+TR^N[lnA[]\SJ[fW!\+TReJLXNjcMU\S[lXUW
\iX\+TN[fJh+QSXUJZJvWUMPbHR3Qix
G
h+QSXUJZJ
#
$
À
R/R
X/r%W
$ G
hiM)HR
$ \S[fbRJ
[ kUR
X/r%W
X/r%W
$ G À
NK MPO/QSR
$ \S[fbRJ
G XNMPQ
h+QSXUJvJ
..............................................................................................................................................
.....
.....
.....
.....
....
....
....
....
...
...
...
...
....
....
....
....
....
....
....
...
............................................................................................................................................
.....
.....
.....
.....
...
...
...
...
....
....
....
....
....
....
....
....
....
....
....
...
..............................................................................................................................................
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
...
...
...
..
...............................................................................................................................................
1
3
4
2
/
BU*3}p~7Dw*46!3=~/(a=#BU70Lf}fU.17S~U2P0At7lPB3=/@*(v2tUw301(p.v796?Pl(i@DÆ38P=3@/Ä}D=4=3@=3
-BÄ#$
O/h+QSXUJZJ#O/QSR{QSRjfO\SR/^:gPr<RhSO/Wr%QS[]\iRR+|Nu
/R hSO3MPJZR
O/h+QSXUJZJAg ^X/r%W?gtO3WP^
$d
`PQSRJZJ+[lXUWJ-Y9XUQ?\+TRJLRqkNO3jcM)R3J-MP Jv[fWPn'O'kNO/Qa[DO/HUjDR x TN MPJ1g[fY/r<R'jlRA\ $!^d X/r%WHRR/KNMPO3j
Oh+QSXUJvJ[DJ x/4 O3 WP^
Oh+QSXUJvJ[DJ 5x TN[fJVOjDJZ$ X\iR1jfjfJ
\iX x g)\+TR3Wr<R WPX/r \+TUO1\
O/h+QSXUJZJ'bMPJ+\sHR%O3W[DWN\SR/nRQ $ G
MPJs\+TUO1À \ x b MP Ji\sHR<^U[ kU[fJv[DHUjDR%H3I 4 gHPR3hiOMPJLR
O/h+QSXUJZJ[DJeOhiM)HPRO/WP^[fJ#XUWUj I5XUWPR{^U[DnA[c\LgP[]\'bMPJi\eHR 1 XUQ 8 $ jD JZX:g
[DWPhiR
Jv[fWPh+R
^X/r%W![fJ ^U[ kU[fJv[DHUjDReH3I 4 g1\+TReXUWUj I!`PXUJZJ+[lHN[fjc[c\S[DR3JO/QSR 12 g 16 g 84 g/XUQ 88
^X/r%MUW \ $
r<
Oh+QSXUJvJebMPJi\HPR 5 \S[fbRJsr<TU O\SRLkUR3QkNOjfM)RVr<R%MPJLR%$Y9XUG Q
d R /WPX/r \+TUO1\
TR<XUWUj]IhiO3JLR#\+TUO
\ \a J'[fJ 84 gr<TN[DhvTbO /R3J
O/h+QSXUJZJR/KN MPO3jP\SX 420 jfJLX:gNJv[fWPh+R $
r<R!WPX/' r /WPX/r\+TUO1\
^X/r%W[DJ 84 g/r<R!O3 jf JLX WPX/r \+TUO\ O/h+QSXUJZJq[DJ 14 (84) = 21
X/rÈ\+TR{XUWUj]IkNOjfM)R!r<R{WPR3R/^[fJ
^PX/r%W?g)O/WP^r<R /$qWPd X/rÈ\+TUO\'[]\bMPJi\eHPR
O%\+TUQSR3R/uv^U[DnA[c\sJLKNMP$ O/QSRNgPJi\SO3Qa\S[fWPnr![c\+T 1 O3WP^RWP^U[fWPn#r![c\+T 0 TRXUWUj]IkNOjfM)R!\+TUO1\
r![fjcjUr<XUQ %[fJ 100
$
...................................................................................................................
.
.
.
.
....
.... 1
.... 2
....
...
...
..
..
d ywWJ+MPbbO/QwI:g/r<R<nRA\F\+TR%JZXNjfMU\S[DXUWXUW{\+TR!QS[DnT\
....
... 2 .... 1 ....
........................................................................................................................
TNMPJAg\+TR!J+MPb XNYP\+TR%^U[DnA[c\aJq\+TUO1\JLXNj kUR#\+TR!h+QSXUJZJZu
...
...
...
...
... 3
...
...
...
....
...
....
...
WUMPbHR3Q[DJ
.... 8 .....
.... 0 .....
.....................................................................................................................
...
...
...
...
$
... 4
...
...
...
...
...
...
...
4
2
0
2 + 1 + 8 + 4 + 2 + 0 + 0 = 17
Å2 ' X\iR\+TUO\VO3jcjFWUMPbHR3QaJ%[DW\+TR`M
ijDR5hiXNMjl^HPRvRQSX O5bM)hvTjDR3JvJ
[DWN\SR3QSRJ+\S[fWPn!JZXNjfMU\S[DXUW U aai]++ V¡v¢sU £A¤i¥L¦:§i¨D¤L e©v©saªl«Z a¥ª ¦t¬Z®+ª]¯1L +° aª ¦-±L¤Sca¥1¡v¤«S§i¦-²]¥1 +¥a9° ¤S³¤i¥ ....
....
....
....
.
.
.
.
...................................................................................................................
´L¡a¨ª µ ° ° ¥w¶+° ¦·P«Lª ¨wa¸+¥ª ¦N¹Pº3¾
#V$'&)(+*,-*/.103254/6798:0<;>=638?0A@CB7D=3EF$
[ kUR3W\+TUO\ ABCDEF [fJOQSR/nAMjfO/QeTR+|3OnXUW5O/WP^ G [fJ\+TR{b[D^Pu+`)XN[fWN\eXNY
AB gU^PRA\iRQib[DWPR\+T
$ R!QaO1\S[lX{XNY\+TR#\SX\SOjPO3QSR3O%XNY:TRv|3O/nXUW ABCDEF \iX%\+TR!O3QSR3O
XNY)\iQS[fO/WPnAjDR GDE
$)BN*3}a~U7D*_46Át=4(aw0}-Ât(al@ÃU01(S)Æ/(a=U2P0
.7S~N2P01?7l~U0A?*/.qÄD(v0.ZÄs(pÆU0At7DDP=$
Ç
1R \ r O3WPd ^ a HR\+TRVQaO^N[fMPJsO3WP^\+TRVO3`)X\+TRb NX Y ABCDEF
QSR3Jv`)R3hS\S[]kUR1j]I
$qd
√
G
%
`P`Uj]IN[DWPn#\+TR TR/UX QSRbCXNY IN\+TUO/nXUQaO/JAgr<R<XHN\aO3[fW a = 23 r TR3W\+TR<O3QSR3OXNY
ABCDEF [fJ
√
3 3 r2 $
6r · a
m o
[ABCDEF ] =
=
2
2
À
[DWPhiRV\+TR%Ojc\S[]\SM)^PR%XNY 4GDE [fJ 2a g\+TR%O3QSR3O<XNY 4GDE [fJ
√
r · 2a
3 2$
[GDE] =
=
r
m o
QSXUb
2
2
m oO/WP^m oigr<R%TUOAkUR
[ABCDEF ]
=
[GDE]
√
3 r2
2
√
3 2
r
2
3
= 3
$
$
$)BU*3}p~7Dw*4/6-*P401(L7FPf}clU.LÃ/f ~U79(v0A@_*t7lt$
#
R nA[ kUReO`QSXXNYr![]\+TXNMU\:r<XUQS^JJLTX/r![fWPns\+TUO1\
$
O/QSRO/Js[DW{\+TR%^U[fOn1QaO3bJ'O/QSR%R/KNMPO3j o
A
F
G
B
...........................................................
.
..
... ..
.. ...
.. ...
.. ..
.. ..... .... .... ... ....
.
.
...
... .. .. ..
...
...
.
.
.
.
...
..
...
...
.... ...
...
...
.. .. ... ...
...................................................................................
.. ...
...
.
.
.
..
.
. .. .. ..
...
.
.
.
.
.
.
...
.
... ...
.
... ..
...
... ....
...
.
... ......
.
.
...... .
... .....
...... ...
... .....
.............................................................
E
C
D
$
[ABCDEF ]
=3
[GDE]
........................................................
.........
. ..
.. ... ..
.. .....
.. .... .....
.
...
..
.
...
..
... .... .....
.
.
...
... ...
...
.....
...
... ..
..
..
.
.
.
.
.
..................................................................................
...
. ...
.
..
.
.
...
. ..
..
.
.
.
.
.
.
...
... ...
...
... ... ...
...
...
... ... ..
...
... ....
... ... ..
... ..
.......
.
.
.........................................................
À
m UT O^R/^
.......................................................
.
... ..
.. ...
.. ...
.. .....
.. ....
.
.
...
..
...
.
...
.
.
..
...
.. ....
...
...
.. ..
..
...
.............................................................................................
. ...
...
.
.
..
...
.. ...
.
.
.
...
...
.
..
...
...
..
...
.
.
...
.
.
...
... ...
... ....
... ..
.........................................................
U a-aici+ q¡v¢:U £A¤i¥L¦1§i¨D¤L -©+©taªD«L a¥1ª ¦¬Z®+ª]¯1L +° aª ¦/±L¤i]a¥¡v¤L«i§S¦²]¥1 +¥a9° ¤A¾
#V$'&)(+*,-*/.103254/6 %030A<3~/(aa
=3Æ3(v0/4/F(v*=U7Dl=$
d
$
WR3O/hvTXNY:[]\SJ mn h+R1$ jfjfJs \+TR
TR 3[DWPn#[DJXUWO/WXU`)RW m × n hvTRJZJZHPXUO3QS^
\iX\aO3jWUMPbHPRQVXNY`PXUJZJ+[lHNjlRbXkUR3J%H3$ I5\+TR 3[DWPn{YlQSXUb \+TUO\#h+R1jfj-[fJr%QS[]\a\iRW
[fWP^
\+TR%JvMPb XNY?Ojfj\+TRJLR mn WUMPbHR3QaJ
BU*3}p~7Dw*4/6Á0A?0/30 30{3=}w*?2P0;>=3.v.10A6
$
G
/[fWPn%$+hS# O/WbXkUR<XUWPR!JZ`OhiR![fWO3WI^N[DQSR3hS\S[lXUWm9TXUQa[ vXUWN\aO3jcj]I:g/kURQa\S[DhiOjfj I?gXUQ
^U[fOnXUWOjfj Io
R!nR1\ 4 hSO/JZR3J :[fY\+TR$3[DWPn[fJq[fW\+TRV[fWN\iRQS[DXUQXNYP\+TR!HPXUO3QS^?gUTR!TUO/J
8 `PXUJZJ+[lHNjlRbXkUR3JmDObXkUR'\iX%R3O/hvT<XNY\+TR 8 J+MPQiQSXNMPWP^N[DWPnJLKNMPO/QSRJioi3[cYPTR[fJXUW<O3W
R/^nR!HUMU\FWPX\[fW{O!h+XUQSWP$R3d QZgUTRVTUO/J 5 `)XUJvJv[DHUjDR#bXkUR3JA/O/WP^<[fY:TR[DJF[DW{O!hiXUQiWPRQLgNTR
TUO/J 3 `)XUJvJv[DHUjDR%bXkUR3J
TNMPJAg[cY)r<R<`UjDO/h+RV\+TR<WUMPbHR3Q'XNYt`PXUJZJ+[lHNjlR%bX3kURJYlQSXUb
R3O/hvT`PXUJv[]\S[lXUW{[fWN\iX\+TUO1\q`)XUJ+[c\S[DXUW?gXNMPQqHPXUO3QS^jDX(
X Jqjf[ /RV\+TN[fJ n h+XNjcMPbWJ
m QSX/r%J
d
z
3
5$
5
8$
5
3
8
5
$$
$$
}|
...
...
$ $
$
...
...
5
8$
3
5$
8
5
5
3
$$
{
$$
TRV\SX\SOj:XNY?O3jcj\+TR%WUMPbHPRQaJeXUW{\+TR<HXUO/QS^[fJ
$
8(m − 2)(n − 2) + 5 2(m − 2) + 2(n − 2) + 3(4) = 8mn − 6m − 6n + 4
N· ¥1:° ¥9+¨l¨w9ªaif«Zª °]+¥)¤+P¨wlp°ci+ 1¾
#$'&)(+*,-*/.103254/6BN0A6=/@=NSÃ!L= l=/(af:=/:2N=/(9@=/@Ât8:*0A.17l=3w(a=/t$
ywW\+TR!JZ`U[DQaOj:HPR1jlX/r `QSX3kUR#\+TUO\
X02 + X12 + X22 + · · · + Xn2 = X02 · X12 · X22 · · · Xn2 g
r<TR3QSRV\+TR%TR[DnT\'XNY?R3O/hvT\iQS[fO/WPnAjDR![fWP^U[DhiO1\iR3^[DW{\+TR%^U[fOn1QaO3b
[DJ 1 MPWU[c\
$
X5
X4
........
..............
...........
.............
1
...........
............
1
...........
...........
............
.........
....
......... ...
.......... ...
......... .... ....
.........
.
.
.
.
.
.
.
.
.
.
.
... .....
.......
... ...
.........
.........
... ....
.........
.
.
.
.
.
.
.
.
.
.......
.......
.....
.........
.
.........
...........................
.........
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.. ...
...............
.
......
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.. ....
...... ..........................................
.
.
.
.
.
.
.
.
.
.
.. ...
..... .............
... ..
...................................................
........................
.. ...
........................
... ...
........................
. ..
........................
........................ ........
.. ...............
... ...
... ...
.
... .
.. ....
... ..
........
....
...
.............
........
X6
....
.. .....
... ......
...
........
3
.... ................ ......
........ ....
..
............
..........
....
.
.
.
...
.
... ..
..
.... ....
.
......
...
.... ....... ....
.
.
2
.
....
.... ..
.....
.... ...
..
.....
.
.
.
.
.
.
.. ..
.... ...
...
.
.
.
.
.
.
.
...
.
......
.......
.....
..
....
...
...........
....
.......... .....
....
...
.......... ....
.
.....
.
.
.
.
.
.
.
.
.
.. ..
.
.
.
.. ....
.....
....
.. ..
..........
..
.....
... ..
..........
... .......
.........
.. ...
1
.... ........ ...................
... ..
. .
...
.. ...
.. ..... ..............
.....
....
.........................
.
.
.
.
.
......................................................................................................................................................................
... ...........................................................................................................................................
...
....... ......
....
X
.............
..........
1
.............
..........
O
1
.....
....... .......
..
X
1
...........
............
q
X
........
..............
q
q
1
X0
BU*3}p~7Dw*_4/6Á~3.17l=N3*Ât(aD@Ã0A(S'tUw301(p.vw2N=U2/ÄÅ)?Å~U0A?*/.FÄl(+0A.vUÄs(aÆN01?7Dl=3
@_*2/?0/246798?0%0/2/7*(p.3$
2
Yn+1
= Xn2 + Yn2
...
.......... .....
...
..
... ....
........
.. ...
..........
.
.
.
.
.
.
.
.
.
.
... ....
......
.
.
.
.
.
.
.
.
.
.. ...
.........
.. ...
..........
.....
..........
......................................................................................................................................
Yn+1............................................
.
.............
.......
1
Xn
............
...........
XUQ n = 1 g 2 g 3 g . . . gq\+TR n\iQa[DO3WPnAjlR
[DW
\+TRJv`[fQaO3jq[DJOQa$ [lnTÇ \!\iQS[fO/WPnAjDR_[DWr<TN[lhvTXUWPR
jlR3nTUO3J{jlRWPn\+$tT d Xn
R1\ Yn HR5\+TR_jlRWPn\+TXNY
\+TReX\+
T$ ' R3Q?jDR/n
TReT/I`PX\iRWU$ MPJLRs\+TR3W<TUO3J-jDR3WPn\+T
Yn+1
X\SRV\+TUO1\ Y = X
I{\+TR % IN\+TUO/n1XUQSR3O3W 0d TR3XUQSR3b_g
Yn
$
m o
W\+TRX\+TR3Q'TUO3WP^?g)H3Ir%Qa[c\S[fWPnV\+TR<O3QSR3O{XNY:\+TR%\iQa[DO3WPnAjlR%[fW\pr<X^N[ tR3QSRWN\r%O1IUJ
mDMPJ+[DWPn\pr<X^N[ tR3QSRWN\'HUO/JZR3J'O3WP^h+XUQSQSR3Jv`)XUWP^N[DWPn!TR[DnT\SJSo+g/r<R%nR1\
$
Yn+1 = Xn Yn
m o
G
`P`Uj]IN[DWPnbO\+TRbO1\S[lhSO3j[fWP^UM)hS\S[DXUW!\iX%m o+g1r![]\+T Y1 = X0 g[c\-hSO/WHRJZTX/r%W
\+TUO\
$
À
d
2
Yn+1
= X02 + X12 + X22 + · · · + Xn2
[Db[fjfO/Qaj]I:gUO3`P`Uj]IN[DWPn#[DWP^NM)hS\S[lXUW{\iXm o+g/r<R<nRA\
Yn+1 = X0 · X1 · X2 · · · Xn
$
TRQSRY9XUQSRNg
2
X02 + X12 + · · · + Xn2 = Yn+1
= X02 · X12 · X22 · · · Xn2
$
U a-aici+ q¡v¢F´L¡a¨ª µ ° ° ¥w¶+° ¦·P«Lª ¨wa¸+¥ª ¦N¹Pº3¾
#V$'&)(+*,-*/.103254/6798:0<;>=638?0A@CB7D=3EF$
d
TR\pr<R3WN\wIPuS\+TN[DQS^>
$ \i RQib XNY#O3WO/Qa[c\+TUbR1\S[DhJZR/KNM)R3WPhiR[fJ\+TUQSR/R\S[DbR3J\+TR
kNO3jcM)RXNY-\+TR Yf\+T\iRQib $ [DWP^5\+TRQaO\S[DXXNY-\+TR{\pr<RWN\Iui\+TN[fQS^5\iRQibC\iX\+TR )QaJi\
\iRQibCXNY:\+$ TR<JLR3KUM)RWPh+R
|`QSR3JvJ'\+TR<QaO1\S[lX[fW\+TR%Y9XUQib p : q r<TR3QSR p O/WP^ q O/QSR
[DWN\SR/nRQaJ
BU*3}p~7Dw*4/6Át=U4(a0L}Ât(aD@Ã0A(S:Æ/(a=U2P0e.17S~U2P0At7l~U0A?*/.Äl(+0A.vÄs(aÆN01?7Dl=P$
Ç
$
R1\ {an } HPRV\+TR!JLR3KUM)RWPh+R![fW{\+TR%`QSXHUjDR3b_g/r<TR3QSR an = a1 + (n − 1)d
À
[DWPhiR a23 = 3 · a5 g/r<R<TUOAkUR
22d + a1
10d
5d
d
TRW
= 3(4d + a1 ) = 12d + 3a1 g
= 2a1$ g
= a1
22d + a1
22d + 5d
27 $
a23
=
=
=
a1
a1
5d
5
U aai]++ V¡v¢sU £A¤i¥L¦:§i¨D¤L e©v©saªl«Z a¥ª ¦t¬Z®+ª]¯1L +° aª ¦-±L¤Sca¥1¡v¤«S§i¦-²]¥1 +¥a9° ¤S³¤i¥ ´L¡a¨ª µ ° ° ¥w¶+° ¦·P«Lª ¨wa¸+¥ª ¦N¹Pº3¾
#!"%$'&)(+*,-*/.103254/6798:0<;>=638?0A@CB7D=3EF$
G
`PXNj]IUWPXUb[fO3j f (x)
$ JvO\S[fJ )R3J#\+TRh+XUWP^N[c\S[DXUW\+TUO1\ f (5 − x) = f (5 + x)
Y9XUQsOjfj:QSRO3j:WU$ MPbHR3QaJ x y9Y f (x) = 0 TUO3J 4 ^N[DJi\S[DWPhS\sQSRO3j:QSXX\aJ1g PWP^\+TR%JvMPb XNY
\+TR3JZR%QSXX\aJ
BU*3}p~7Dw*{4/6Ä} P=/UNÆ/(a=U2P
0 .17S~U2P0At7lUBU; ;0798:*2/D.17l:&U=1}0At4=~3Æ 9?2P*:0A.+l=$
À
X /[fWPnV$Od \F\+TR
[DWPhiR f (x) = 0 TUO3J 4 ^U[fJ+\S[fWPhS\qQSR3Oj:QSXX\aJ'O/WP^r<R!O3QSR<XUWUj]I{jDX(
JvMPb XNY\+TR3JZRVQSXX\aJ1gr<RVbO1I{O3JZJ+MPbRe\+TUO\ f (x) [fJsO%KNMPO/Qp\S[lh`)XNj IWPXUb[DOj TNMPJ1g
f (x) = ax4 + bx3 + cx2 + dx + e g
O/WP^{\+TR!JvMPb
f (5 − x)
f (5 + x)
XNY)\+TR%QSXX\aJXNY f (x) [DJ −b/a
$
'
X/r
= ax4 − x3 (20a + b) + x2 (150a + 15b + c)
− x(500a + 75b + 10c + d) + 625a + 125b + 25c + 5d + e g
= ax4 + x3 (20a + b) + x2 (150a + 15b + c)
$
+ x(500a + 75b + 10c + d) + 625a + 125b + 25c + 5d + e
$d
RWPh+RNg 20a$ + b = 0 /\+TUO\[fJ1g b/a = −20 TR3QSR1Y9XUQSRUg\+TR%JvMPb XNY)\+TR%QSXX\aJXNY
f (x) [fJ 20
Å2 ' X\iRV\+TUO\ f [fJeJaIUbbR1\iQa[lh'O/HPXNMU\ x = 5 N\+TRQSRY9XUQSRNgU\+TR<JvMPb XNY?\+TR
$
Y9XNMPQqQSR3Oj)QSXX\SJ'bMPJ+\sHR 4 × 5 = 20 U a-aici+ q¡v¢»N«+aª ¤Liq¼¨D° ¸q¶La¨D¦P®¥L°]+a¨D9°] v¤L qº+:½Aº1½v¦ µ «La¥1+NU° ¨wa9¦L¨f§Za¥ª ° ¥L¤A¾
#<¿V$'&)(+*,-*/.103254/6798:0<;>=638?0A@CB7D=3EF$
$ G
`PR3QaJZXUW5hiO3JLTRJeO{hvTR/KNM)RO\\+TR{HUO/W
Ib[fJ+\aO /RNgU\+TR!\iR1$
jfjDdR3Qs`O1IUJ\+TR
WUMPbHR3Q#XNYFh+RWN\SJVO/JV^PXNjcjDO3QaJVO/WP^\+TRWUMPbHR3Q#XNYF^PXNjcjDO3QaJVO3J!h+RWN\SJ
TR`)RQaJLXUW
JZ`PR3WP^UJ $3.50 HRY9XUQSRWPX\S[Dhi[DWPn\+TRb[fJ+\aO /RNg1\+TRW?g/XUW<hiXNMPWN\S[DWPnq\+TRbXUWPRLI?g $ PWP ^J
\+TUO\s\+TR{QSR3bO3[fWU[DWPnbXUWPRLI[DJ#R+|3OhS\Sj I_^XNM)HUjDR<\+TR{O/bXNMPWN\XNYt\+TRhvTR/KNM)R
XUQ
r<TUO\qO3bXNMPWN\r%O/Jq\+TR%hvTR/KNM)R%bO/^PR%XNMU\+x
BU*3}p~7Dw*4/6Á0A?0/30 30{3=}w*?2P0;>=3.v.10A6$
Ç
R1$
\ d d O3WP^ c QSR`PQSRJLRWN\\+TR{WUMPbHR3QXNY^PXNjcjDO3QaJ#O/WP^hiR3WN\aJ!XUW\+$
TRÀ XUQS[DnA[DWOj
hvTR/KNM)R
TNMPJ1g\+TRO3bXNMPWN\!XNYF\+TR5hvTR/KNM)RUg-[DW h+RWN\SJAg-[DJ 100d + c [fWPh+${Rd r<R
O3j r%O1IUJ!TUO1kUR{jlRJZJ\+TUO3W 100 h+RWN\SJ!XUWOhvTR/KNM)RUg:r<RbMPJ+\#TUOAkUR c < 100 TR
[DWUY9XUQSbO1\S[lXUW[fW{\+TR%`QSXHUjDR3b IN[lR1jl^UJ
d
À
TRW
100c + d − 350
98c − 350
14(7c − 25)
= 2(100d + c) g
= 199d g$
= 199d
[DWPhiR 199 [fJs`PQa[DbRUg3r<RVbMPJi\qTUO1kUR d = 14k Y9XUQFJLXUbRV`)XUJ+[c\S[ kUR#[DWN\SR/nRQ k
$
7c − 25 = 199k
$
$ #
9y Y k = 1 g)r<RnRA\ c = 32 O3WP^ d = 14
TR3Wr<R{\iQwI k = 2 O3WP^ k = 3 g:r<R
$ ^PXWPX\nRA\O/W[DWN\SR/nRQ
L
J
N
X
c
j
U
M
S
\
l
[
U
X
W
9
Y
U
X
Q
c
D
[
n
T
R3QqkNOjfM)RJVXNY k r![fjcj?Y9XUQSh+R $ c > 100 g
$-d
r<TN[lhvT{[DJ'WPX\O3jcjlX/r<R3^
TNMPJ1g\+TR!O/bXNMPWN\'XUW\+TR<hvTR/KNM)R#r%O/J 14.32
# %$&)(v*,-*/.A0/2!4/6FÄs:2N(v0 >;>=U*/)Át(S=N2P
0 s.17S~U2P0At7lÄ 9 ~Na=.BN0/i*?2N=3(i6
BUS8:**3}DN*?2P*U $
$
-XNMO/QSR!nA[ kUR3WOVJLTR3R1\XNY:`O/`PR3QXNY)Jv[ vR 2003 × 2004 -XNMO3QSR!$OjfjDX/r<R/^%\SX
hiMU\[c\qR[]\+TR3QqTXUQS[ vXUWN\SOjfj IXUQ-kURQa\S[DhiOjfj I$ mf\+TUO1\[DJAgN`PO3QaO3jcjlR1j\iX{O/WR3^PnRo -XNMr![DJZT
\iXXHN\aO3[fW 2003 × 2004
$ MPWU[c\VJLKNMPO/QSRJ -XNMO/QSRWPX\VO3jcjlX/r<R3^\SXY9XNjl^ XUQ#$ J+\aOh `[DR/hiR3JXNY)\+TR%`O/`PR3Q
RA\iRQib[DWPR#\+TR%b[DWU[fbMPb WUMPbHPRQ'XNY?hiMU\SJ'QSR/KNM[DQSR3^
ÅP
2 %*<.A*3}a~U7D*U.e8=U3040301(+0/i0w30/
2 38:0e,?(+*P43}0A@ (v0A@=D.#*,-01
© Copyright 2026 Paperzz