class 11th maths chapter 9

7/23/2019 Class 11th Maths Chapter 9

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Noturoh nug̀ crs orc tbc preduit ea bugon spmrmt. ’ DCDCLMND

=.? Mntreduitmen

Mn gotbcgotmis( tbc werd( ‘ scqucnic‗ ms uscd mn guib tbc

sogc woy os mt ms mn erdmnory Cnjhmsb. Zbcn wc soy tbot o

iehhcitmen ea e`kcits ms hmstcd mn o scqucnic( wc usuohhy gcon

tbot tbc iehhcitmen ms erdcrcd mn suib o woy tbot mt bos on

mdcntmamcd amrst gcg`cr( sciend gcg`cr( tbmrd gcg`cr ond

se en. Aer cxogphc( pepuhotmen ea bugon `cmnjs er ̀ oitcrmo

ot dmaacrcnt tmgcs aerg o scqucnic. Tbc ogeunt ea gency

dcpesmtcd mn o ̀ onl( evcr o nug`cr ea ycors aerg o scqucnic.

Dcprcimotcd vohucs ea icrtomn ieggedmty eiiur mn oscqucnic. Ycqucnics bovc mgpertont opphmiotmens mn scvcroh

spbcrcs ea bugon oitmvmtmcs.

Ycqucnics( aehhewmnj spcimami pottcrns orc iohhcd prejrcssmens. Mn prcvmeus ihoss(

wc bovc studmcd oèut ormtbgctmi prejrcssmen "O.V-. Mn tbms Iboptcr( `csmdcs dmsiussmnj

gerc oèut O.V.< ormtbgctmi gcon( jcegctrmi gcon( rchotmensbmp `ctwccn O.G.

ond J.G.( spcimoh scrmcs mn aergs ea sug te n tcrgs ea iensciutmvc noturoh nug`crs(

sug te n tcrgs ea squorcs ea noturoh nug`crs ond sug te n tcrgs ea iu`cs ea

noturoh nug`crs wmhh ohse `c studmcd.

=.0 Ycqucnics

Hct us iensmdcr tbc aehhewmnj cxogphcs;

Ossugc tbot tbcrc ms o jcncrotmen jop ea 96 ycors( wc orc oslcd te amnd tbc

nug`cr ea onicsters( m.c.( porcnts( jrondporcnts( jrcot jrondporcnts( cti. tbot o pcrsen

gmjbt bovc evcr 966 ycors.

Bcrc( tbc tetoh nug`cr ea jcncrotmens 7966

?696

7

Amènoiim

"??:3'?036-

Iboptcr

YC_SCNICY OND YCPMCY

=



?:2 GOTBCGOTMIY

Tbc nug`cr ea pcrsen”s onicsters aer tbc amrst( sciend( tbmrd( …( tcntb jcncrotmens orc

0( 4( 2( ?1( 90( …( ?604. Tbcsc nug`crs aerg wbot wc iohh o scqucnic.

Iensmdcr tbc suiicssmvc quetmcnts tbot wc e`tomn mn tbc dmvmsmen ea ?6 `y 9 ot

dmaacrcnt stcps ea dmvmsmen. Mn tbms preicss wc jct 9(9.9(9.99(9.999( ... ond se en. Tbcsc

quetmcnts ohse aerg o scqucnic. Tbc vormeus nug`crs eiiurrmnj mn o scqucnic orc

iohhcd mts tcrgs. Zc dcnetc tbc tcrgs ea o scqucnic `y o? ( o

0 ( o

9 ( …( o

n( …( cti.( tbc

su`sirmpts dcnetc tbc pesmtmen ea tbc tcrg. Tbc ntb tcrg ms tbc nug`cr ot tbc ntb pesmtmen

ea tbc scqucnic ond ms dcnetcd ̀ y on.

Tbc ntb tcrg ms ohse iohhcd tbc jcncroh

tcrg ea tbc

scqucnic.

Tbus( tbc tcrgs ea tbc scqucnic ea pcrsen”s onicsters gcntmencd oèvc orc;o

?7 0 ( o

07 4 ( o

97 2 ( …( o

?67 ?604.

Ymgmhorhy( mn tbc cxogphc ea suiicssmvc quetmcnts

o?

7 9( o0

7 9.9( o9

7 9.99( …( o1

7 9.99999( cti.

O scqucnic ientomnmnj amnmtc nug`cr ea tcrgs ms iohhcd o amnmtc scqucnic. Aer

cxogphc( scqucnic ea onicsters ms o amnmtc scqucnic smnic mt ientomns ?6 tcrgs "o amxcd

nug`cr-.

O scqucnic ms iohhcd mnamnmtc( ma mt ms net o amnmtc scqucnic. Aer cxogphc( tbc

scqucnic ea suiicssmvc quetmcnts gcntmencd oèvc ms on mnamnmtc scqucnic( mnamnmtc mn

tbc scnsc tbot mt ncvcr cnds.

Eatcn( mt ms pessm`hc te cxprcss tbc ruhc( wbmib ymchds tbc vormeus tcrgs ea o scqucnic

mn tcrgs ea ohjc`romi aerguho. Iensmdcr aer mnstonic( tbc scqucnic ea cvcn noturohnug`crs 0( 4( 1( …

Bcrc o?

7 0 7 0 Ú ? o0

7 4 7 0 Ú 0

o9

7 1 7 0 Ú 9 o4

7 2 7 0 Ú 4

.... . ... .... .... . ... ....

.... . ... .... .... . ... ....

o09

7 41 7 0 Ú 09( o04

7 42 7 0 Ú 04( ond se en.

Mn aoit( wc scc tbot tbc ntb tcrg ea tbms scqucnic ion `c wrmttcn os on

7

0n(

wbcrc n ms o noturoh nug`cr. Ymgmhorhy( mn tbc scqucnic ea edd noturoh nug`crs ?(9(3( …(

tbc ntb

tcrg ms jmvcn `y tbc aerguho( on 7 0n ’ ?( wbcrc n ms o noturoh nug`cr.Mn segc ioscs( on orronjcgcnt ea nug`crs suib os ?( ?( 0( 9( 3( 2(.. bos ne vmsm`hc

pottcrn( ùt tbc scqucnic ms jcncrotcd `y tbc rciurrcnic rchotmen jmvcn `y

o?

7 o0

7 ?

o9

7 o?

, o0

on

7 on ’ 0

, on ’ ?

( n > 0

Tbms scqucnic ms iohhcd Amènoiim scqucnic.



YC_SCNICY OND YCPMCY ?:=

Mn tbc scqucnic ea prmgcs 0(9(3(:(…( wc amnd tbot tbcrc ms ne aerguho aer tbc ntb

prmgc. Yuib scqucnic ion enhy `c dcsirm`cd `y vcr`oh dcsirmptmen.

Mn cvcry scqucnic( wc sbeuhd net cxpcit tbot mts tcrgs wmhh ncicssormhy `c jmvcn

`y o spcimami aerguho. Bewcvcr( wc cxpcit o tbcerctmioh sibcgc er o ruhc aer jcncrotmnj

tbc tcrgs o?( o

0( o

9(…(o

n(… mn suiicssmen.

Mn vmcw ea tbc o`evc( o scqucnic ion `c rcjordcd os o aunitmen wbesc degomn

ms tbc sct ea noturoh nug`crs er segc su`sct ea mt ea tbc typc {?( 0( 9...l }. Yegctmgcs(

wc usc tbc aunitmenoh netotmen o"n- aer on.

=.9 YcrmcsHct o

?( o

0( o

9(…(o

n( `c o jmvcn scqucnic. Tbcn( tbc cxprcssmen

o?

, o0,

o

9,(…, o

n,

...

ms iohhcd tbc scrmcs osseimotcd wmtb tbc jmvcn scqucnic .Tbc scrmcs ms amnmtc er mnamnmtc

oiierdmnj os tbc jmvcn scqucnic ms amnmtc er mnamnmtc. Ycrmcs orc eatcn rcprcscntcd mn

iegpoit aerg( iohhcd smjgo netotmen( usmnj tbc Jrccl hcttcr ∟ "smjgo- os gcons ea

mndmiotmnj tbc suggotmen mnvehvcd. Tbus( tbc scrmcs o? , o0 , o9 , ... , on ms o``rcvmotcd

os

?

n

l

l

o7

∟.

Pcgorl Zbcn tbc scrmcs ms uscd( mt rcacrs te tbc mndmiotcd sug net te tbc sug mtscha.

Aer cxogphc( ? , 9 , 3 , : ms o amnmtc scrmcs wmtb aeur tcrgs. Zbcn wc usc tbc pbrosc

‘ sug ea o scrmcs(‗ wc wmhh gcon tbc nug`cr tbot rcsuhts areg oddmnj tbc tcrgs( tbc

sug ea tbc scrmcs ms ?1.

Zc new iensmdcr segc cxogphcs.

Cxogphc ? Zrmtc tbc amrst tbrcc tcrgs mn coib ea tbc aehhewmnj scqucnics dcamncd `y

tbc aehhewmnj;

"m- on 7 0n , 3( "mm- on 79

4

n ∐.

Yehutmen "m- Bcrc on 7 0n , 3

Yu`stmtutmnjn 7 ?( 0( 9( wc jct

o?

7 0"?- , 3 7 :( o0

7 =( o9

7 ??

Tbcrcaerc( tbc rcqumrcd tcrgs orc :( = ond ??.

"mm- Bcrc on 7

9

4

n ∐. Tbus( ? 0 9

? 9 ? ?6

4 0 4o ( o ( o

∐7 7 ∐ 7 ∐ 7



?26 GOTBCGOTMIY

Bcnic( tbc amrst tbrcc tcrgs orc ? ?

0 4 ’ ( ’ ond 6.

Cxogphc 0 Zbot ms tbc 06tb tcrg ea tbc scqucnic dcamncd `y

on 7 "n ’ ?- "0 ’ n- "9 , n- 5

Yehutmen Vuttmnj n 7 06 ( wc e`tomn

o06

7 "06 ’ ?- "0 ’ 06- "9 , 06-

7 ?= Ú "’ ?2- Ú "09- 7 ’ :211.

Cxogphc 9 Hct tbc scqucnic on `c dcamncd os aehhews;

o?

7 ?( on

7 on ’ ?

, 0 aer n ≣ 0.

Amnd amrst amvc tcrgs ond wrmtc ierrcspendmnj scrmcs.

Yehutmen Zc bovc

o?

7 ?( o0

7 o?

, 0 7 ? , 0 7 9( o9

7 o0

, 0 7 9 , 0 7 3(

o4

7 o9

, 0 7 3 , 0 7 :( o3

7 o4

, 0 7 : , 0 7 =.

Bcnic( tbc amrst amvc tcrgs ea tbc scqucnic orc ?(9(3(: ond =. Tbc ierrcspendmnj scrmcs

ms ? , 9 , 3 , : , = ,...

CRCPIMYC =.?

Zrmtc tbc amrst amvc tcrgs ea coib ea tbc scqucnics mn Cxcrimscs ? te 1 wbesc ntb

tcrgs orc;

?. on

7 n "n , 0- 0. on7

?

n

n , 9. on7 0n

4. on

70 9

1

n ∐3. o

n7 "’?-n ’? 3n,? 1. o

n

0 3

4

nn

,7 .

Amnd tbc mndmiotcd tcrgs mn coib ea tbc scqucnics mn Cxcrimscs : te ?6 wbesc ntb

tcrgs orc;

:. on7 4n ’ 9< o

?:( o

042. o

n7

0

:<0n

no

=. on

7 "’?-n ’ ?n9< o=

?6. 06

" ’ 0-<

9n

n no o

n7

,.



YC_SCNICY OND YCPMCY ?2?

Zrmtc tbc amrst amvc tcrgs ea coib ea tbc scqucnics mn Cxcrimscs ?? te ?9 ond e`tomn tbc

ierrcspendmnj scrmcs;

??. o?

7 9( on

7 9on ’ ?

, 0 aer ohh n > ? ?0. o?

7 ’ ?( on7

?no

n∐

( n ≣ 0

?9. o?

7 o0 7 0( o

n7 o

n ’ ? ’? ( n > 0

?4. Tbc Am`enoiim scqucnic ms dcamncd `y

? 7 o?

7 o0

ond on

7 on ’ ?

, on ’ 0

( n > 0.

Amnd?n

n

o

o

,( aer n 7 ?( 0( 9( 4( 3

=.4 Ormtbgctmi Vrejrcssmen "O.V.-

Hct us rciohh segc aerguhoc ond prepcrtmcs studmcd corhmcr.

O scqucnic o? ( o

0( o

9 (…( o

n (… ms iohhcd ormtbgctmi scqucnic er ormtbgctmi

prejrcssmen ma on , ?

7 on

, d ( n ∂ N( wbcrc o?

ms iohhcd tbc amrst tcrg ond tbc ienstont

tcrg d ms iohhcd tbc ieggen dmaacrcnic ea tbc O.V.

Hct us iensmdcr on O.V. "mn mts stondord aerg- wmtb amrst tcrg o ond ieggen

dmaacrcnic d ( m.c.( o( o , d ( o , 0d ( ...Tbcn tbc ntb tcrg " jcncroh tcrgea tbc O.V. ms o

n7 o , "n ’ ?- d.

Zc ion vcrmay tbc aehhewmnj smgphc prepcrtmcs ea on O.V. ;

"m- Ma o ienstont ms oddcd te coib tcrg ea on O.V.( tbc rcsuhtmnj scqucnic ms

ohse on O.V.

"mm- Ma o ienstont ms su`troitcd areg coib tcrg ea on O.V.( tbc rcsuhtmnj

scqucnic ms ohse on O.V.

"mmm- Ma coib tcrg ea on O.V. ms guhtmphmcd ̀ y o ienstont( tbcn tbc rcsuhtmnj

scqucnic ms ohse on O.V.

"mv- Ma coib tcrg ea on O.V. ms dmvmdcd `y o nen'zcre ienstont tbcn tbc

rcsuhtmnj scqucnic ms ohse on O.V.

Bcrc( wc sbohh usc tbc aehhewmnj netotmens aer on ormtbgctmi prejrcssmen;

o 7 tbc amrst tcrg( h 7 tbc host tcrg( d 7 ieggen dmaacrcnic(

n 7 tbc nug`cr ea tcrgs.

Yn7 tbc sug te n tcrgs ea O.V.

Hct o( o , d( o , 0d( …( o , "n ’ ?- d `c on O.V. Tbcn

h 7 o , "n ’ ?- d



?20 GOTBCGOTMIY

\ WY 0 " ?-0

n

no n d 7 , ∐

Zc ion ohse wrmtc( \ WY0

n

no h 7 ,

Hct us iensmdcr segc cxogphcs.

Cxogphc 4 Mn on O.V. ma gtb

tcrg ms n ond tbc ntb

tcrg ms g( wbcrc g ≦ n( amnd tbc ptbtcrg.

Yehutmen Zc bovc og

7 o , "g ’ ?- d 7 n( ... "?-

ond on

7 o , "n ’ ?- d 7 g ... "0-

Yehvmnj "?- ond "0-( wc jct

"g ’ n- d 7 n ’ g( er d 7 ’ ?( ... "9-

ond o 7 n , g ’ ? ... "4-

Tbcrcaerc o p7 o , " p ’ ?-d

7 n , g ’ ? , " p ’ ?- "’?- 7 n , g ’ p

Bcnic( tbc ptb tcrg ms n , g ’ p.

Cxogphc 3 Ma tbc sug ea n tcrgs ea on O.V. ms ?V " ’?-_0

n n n, ( wbcrc V ond _

orc ienstonts( amnd tbc ieggen dmaacrcnic.

Yehutmen Hct o?( o

0( … o

n `c tbc jmvcn O.V. Tbcn

Yn

7 o?

, o0

, o9

,..., on ’?

, on

7 nV ,?

0n "n ’ ?- _

Tbcrcaerc Y?

7 o?

7 V( Y0

7 o?

, o0

7 0V , _

Ye tbot o0

7 Y0

’ Y?

7 V , _

Bcnic( tbc ieggen dmaacrcnic ms jmvcn `y d 7 o0

’ o?

7 "V , _- ’ V 7 _.

Cxogphc 1 Tbc sug ea n tcrgs ea twe ormtbgctmi prejrcssmens orc mn tbc rotme"9n , 2- ; ":n , ?3-. Amnd tbc rotme ea tbcmr ?0tb tcrgs.

Yehutmen Hct o?( o

0ond d

?( d

0`c tbc amrst tcrgs ond ieggen dmaacrcnic ea tbc amrst

ond sciend ormtbgctmi prejrcssmen( rcspcitmvchy. Oiierdmnj te tbc jmvcn iendmtmen( wc

bovc

9 2

: ?3

Yug te tcrgsea amrst O.V.

Yugte tcrgsea sciendO.V.

n

n

n

n

,

,7



YC_SCNICY OND YCPMCY ?29

er

\ W

\ W

? ?

0 0

0 ?9 20

: ?30 ?

0

no " n -d

n

n no " n -d

, ∐,

7,, ∐

er ? ?

0 0

0 " ?- 9 2

0 " ?- : ?3

o n d n

o n d n

, ∐ ,7

, ∐ , ... "?-

New

tb

? ?tb

0 0

???0 tcrgea amrst O.V.???0 tcrgea sciend O.V

o d o d

,7,

? ?

0 0

0 00 9 09 2

0 00 : 09 ?3

o d

o d

, Ú ,7

, Ú , \@y puttmnj n 7 09 mn "?-W

Tbcrcaerc

tb? ?

tb0 0

?? ?0 tcrg ea amrst O.V. :

?? ?1?0 tcrg ea sciend O.V.

o d

o d

,7 7

,

Bcnic( tbc rcqumrcd rotme ms : ; ?1.

Cxogphc : Tbc mniegc ea o pcrsen ms Ps. 9(66(666( mn tbc amrst ycor ond bc rcicmvcs on

mnircosc ea Ps.?6(666 te bms mniegc pcr ycor aer tbc ncxt ?= ycors. Amnd tbc tetohogeunt( bc rcicmvcd mn 06 ycors.

Yehutmen Bcrc( wc bovc on O.V. wmtb o 7 9(66(666( d 7 ?6(666( ond n 7 06.

Ssmnj tbc sug aerguho( wc jct(

06

06Y \166666 ?= ?6666W

07 , Ú 7 ?6 ":=6666- 7 :=(66(666.

Bcnic( tbc pcrsen rcicmvcd Ps. :=(66(666 os tbc tetoh ogeunt ot tbc cnd ea 06 ycors.

=.4.? Ormtbgctmi gcon Jmvcn twe nug`crs o ond `. Zc ion mnscrt o nug`cr O

`ctwccn tbcg se tbot o( O( ` ms on O.V. Yuib o nug`cr O ms iohhcd tbc ormtbgctmi gcon

"O.G.- ea tbc nug`crs o ond `. Netc tbot( mn tbms iosc( wc bovc

O ’ o 7 ` ’ O( m.c.( O 70

o `,

Zc goy ohse mntcrprct tbc O.G. `ctwccn twe nug`crs o ond ` os tbcmr

ovcrojc0

o `,. Aer cxogphc( tbc O.G. ea twe nug`crs 4 ond ?1 ms ?6. Zc bovc( tbus

ienstruitcd on O.V. 4( ?6( ?1 `y mnscrtmnj o nug`cr ?6 `ctwccn 4 ond ?1. Tbc noturoh



?24 GOTBCGOTMIY

qucstmen new ormscs ; Ion wc mnscrt twe er gerc nug`crs ̀ ctwccn jmvcn twe nug`crs

se tbot tbc rcsuhtmnj scqucnic iegcs eut te `c on O.V. 5 E`scrvc tbot twe nug`crs 2

ond ?0 ion `c mnscrtcd `ctwccn 4 ond ?1 se tbot tbc rcsuhtmnj scqucnic 4( 2( ?0( ?1

`ciegcs on O.V.

Gerc jcncrohhy( jmvcn ony twe nug`crs o ond `( wc ion mnscrt os gony nug`crs

os wc hmlc `ctwccn tbcg suib tbot tbc rcsuhtmnj scqucnic ms on O.V.

Hct O?( O

0( O

9( …( O

n`c n nug`crs `ctwccn o ond ` suib tbot o( O

?( O

0( O

9( …(

On( ` ms on O.V.

Bcrc( ` ms tbc "n , 0-tb

tcrg( m.c.( ` 7 o , \"n , 0- ’ ?Wd 7 o , "n , ?- d.

Tbms jmvcs?

` od

n

∐7

,.

Tbus( n nug`crs `ctwccn o ond ` orc os aehhews;

O?

7 o , d 7 o ,?

` o

n

∐

,

O0

7 o , 0d 7 o ,0" -

?

` o

n

∐

,

O9

7 o , 9d 7 o ,9" -

?` on

∐,

. .. .. ..... . .... .....

. .. .. ..... . .... .....

On

7 o , nd 7 o ," -

?

n ` o

n

∐

, .

Cxogphc 2 Mnscrt 1 nug`crs `ctwccn 9 ond 04 suib tbot tbc rcsuhtmnj scqucnic ms

on O.V.

Yehutmen Hct O?( O

0( O

9( O

4( O

3ond O

1 `c smx nug`crs `ctwccn 9 ond 04 suib tbot

9( O?( O0( O9( O4( O3( O1( 04 orc mn O.V. Bcrc( o 7 9( ` 7 04( n 7 2.

Tbcrcaerc( 04 7 9 , "2 ’?- d ( se tbot d 7 9.

Tbus O?

7 o , d 7 9 , 9 7 1< O0

7 o , 0d 7 9 , 0 Ú 9 7 =<

O9

7 o , 9d 7 9 , 9 Ú 9 7 ?0< O4

7 o , 4d 7 9 , 4 Ú 9 7 ?3<

O3

7 o , 3d 7 9 , 3 Ú 9 7 ?2< O1

7 o , 1d 7 9 , 1 Ú 9 7 0?.

Bcnic( smx nug`crs `ctwccn 9 ond 04 orc 1( =( ?0( ?3( ?2 ond 0?.



YC_SCNICY OND YCPMCY ?23

CRCPIMYC =.0

?. Amnd tbc sug ea edd mntcjcrs areg ? te 066?.

0. Amnd tbc sug ea ohh noturoh nug`crs hymnj ̀ ctwccn ?66 ond ?666( wbmib orc

guhtmphcs ea 3.

9. Mn on O.V.( tbc amrst tcrg ms 0 ond tbc sug ea tbc amrst amvc tcrgs ms enc'aeurtb ea

tbc ncxt amvc tcrgs. Ybew tbot 06 tb tcrg ms ’??0.

4. Bew gony tcrgs ea tbc O.V. ’ 1(??

0∐ ( ’ 3( … orc nccdcd te jmvc tbc sug ’035

3. Mn on O.V.( ma ptb tcrg ms?

qond qtb tcrg ms

?

p( prevc tbot tbc sug ea amrst pq

tcrgs ms?

0" pq ,?-( wbcrc p ≦ q.

1. Ma tbc sug ea o icrtomn nug`cr ea tcrgs ea tbc O.V. 03( 00( ?=( … ms ??1. Amnd tbc

host tcrg.

:. Amnd tbc sug te n tcrgs ea tbc O.V.( wbesc l tb tcrg ms 3l , ?.

2. Ma tbc sug ea n tcrgs ea on O.V. ms " pn , qn0-( wbcrc p ond q orc ienstonts(

amnd tbc ieggen dmaacrcnic.

=. Tbc sugs ea n tcrgs ea twe ormtbgctmi prejrcssmens orc mn tbc rotme3n , 4 ; =n , 1. Amnd tbc rotme ea tbcmr ?2tb tcrgs.

?6 . Ma tbc sug ea amrst p tcrgs ea on O.V. ms cquoh te tbc sug ea tbc amrst q tcrgs( tbcn

amnd tbc sug ea tbc amrst " p , q- tcrgs.

??. Yug ea tbc amrst p( q ond r tcrgs ea on O.V orc. o( ` ond i( rcspcitmvchy.

Vrevc tbot " - " - " - 6o ` i

q r r p p q p q r

∐ , ∐ , ∐ 7

?0. Tbc rotme ea tbc sugs ea g ond n tcrgs ea on O.V. ms g0 ; n0. Ybew tbot tbc rotme

ea gtb ond ntb tcrg ms "0g ’ ?- ; "0n ’ ?-.

?9. Ma tbc sug ea n tcrgs ea on O.V. ms 9n0 , 3n ond mts gtb tcrg ms ?14( amnd tbc vohuc

ea g.?4. Mnscrt amvc nug`crs ̀ ctwccn 2 ond 01 suib tbot tbc rcsuhtmnj scqucnic ms on O.V.

?3. Ma ? ?

n n

n n

o `

o `∐ ∐

,

,ms tbc O.G. `ctwccn o ond `( tbcn amnd tbc vohuc ea n.

?1. @ctwccn ? ond 9?( g nug`crs bovc ̀ ccn mnscrtcd mn suib o woy tbot tbc rcsuhtmnj

scqucnic ms on O. V. ond tbc rotme ea :tb ond "g ’ ?-tb nug`crs ms 3 ; =. Amnd tbc

vohuc ea g.



?21 GOTBCGOTMIY

?:. O gon storts rcpoymnj o heon os amrst mnstohgcnt ea Ps. ?66. Ma bc mnircoscs tbc

mnstohgcnt ̀ y Ps 3 cvcry gentb( wbot ogeunt bc wmhh poy mn tbc 96tb mnstohgcnt5

?2 . Tbc dmaacrcnic `ctwccn ony twe iensciutmvc mntcrmer onjhcs ea o pehyjen ms 3¶.

Ma tbc sgohhcst onjhc ms ?06¶ ( amnd tbc nug`cr ea tbc smdcs ea tbc pehyjen.

=.3 Jcegctrmi Vrejrcssmen "J . V.-

Hct us iensmdcr tbc aehhewmnj scqucnics;

"m- 0(4(2(?1(...( "mm-? ? ? ?

= 0: 2? 049

’ ’ ( ( (

...

"mmm- . (. (. (...6? 666? 66666?

Mn coib ea tbcsc scqucnics( bew tbcmr tcrgs prejrcss5 Zc netc tbot coib tcrg( cxicpt

tbc amrst prejrcsscs mn o dcamnmtc erdcr.

Mn "m-( wc bovc

90 4?

? 0 9

0( 0( 0( 0oo o

o

o o o

7 7 7 7

ond se en.

Mn "mm-( wc e`scrvc( 90 4?

? 0 9

? ? ? ?( ( (

= 9 9 9

oo o ’ ’ ’ o

o o o7 7 7 7 ond se en.

Ymgmhorhy( stotc bew de tbc tcrgs mn "mmm- prejrcss5 Mt ms e`scrvcd tbot mn coib iosc(

cvcry tcrg cxicpt tbc amrst tcrg ̀ cors o ienstont rotme te tbc tcrg mggcdmotchy prcicdmnj

mt. Mn "m-( tbms ienstont rotme ms 0< mn "mm-( mt ms?

9 ’ ond mn "mmm-( tbc ienstont rotme ms 6.6?.

Yuib scqucnics orc iohhcd jcegctrmi scqucnic er jcegctrmi prejrcssmen o``rcvmotcd

os J.V.

O scqucnic o?( o

0( o

9( …( o

n( … ms iohhcd jcegctrmi prejrcssmen( ma coib tcrg ms

nen'zcre ond?l

l

o

o

,7 r "ienstont-( aer l ≣ ?.

@y hcttmnj o?

7 o( wc e`tomn o jcegctrmi prejrcssmen( o( or ( or 0( or 9(….( wbcrc o

ms iohhcd tbc amrst tcrg ond r ms iohhcd tbc ieggen rotme ea tbc J.V. Ieggen rotme mn

jcegctrmi prejrcssmen "m-( "mm- ond "mmm- o`evc orc 0(?

9 ’ ond 6.6?( rcspcitmvchy.

Os mn iosc ea ormtbgctmi prejrcssmen( tbc pre`hcg ea amndmnj tbc ntb tcrg er sug ea n

tcrgs ea o jcegctrmi prejrcssmen ientomnmnj o horjc nug`cr ea tcrgs weuhd ̀ c dmaamiuht

wmtbeut tbc usc ea tbc aerguhoc wbmib wc sbohh dcvchep mn tbc ncxt Ycitmen. Zc sbohh

usc tbc aehhewmnj netotmens wmtb tbcsc aerguhoc;

o 7 tbc amrst tcrg( r 7 tbc ieggen rotme( h 7 tbc host tcrg(



YC_SCNICY OND YCPMCY ?2:

n 7 tbc nug`crs ea tcrgs(

Yn

7 tbc sug ea n tcrgs.

=.3.? Jcncroh tcrg ea o J .V. Hct us iensmdcr o J.V. wmtb amrst nen'zcre tcrg ‒o” ond

ieggen rotme ‒r ”. Zrmtc o acw tcrgs ea mt. Tbc sciend tcrg ms e`tomncd ̀ y guhtmphymnj

o `y r ( tbus o0

7 or . Ymgmhorhy( tbmrd tcrg ms e`tomncd `y guhtmphymnj o0

`y r . Tbus(

o9

7 o0r 7 or 0( ond se en.

Zc wrmtc `chew tbcsc ond acw gerc tcrgs.

?st tcrg 7 o?

7 o 7 or ?’?( 0nd tcrg 7 o0

7 or 7 or 0’?( 9rd tcrg 7 o9

7 or 0 7 or 9’?

4tb tcrg 7 o4 7 or 9 7 or 4’?( 3tb tcrg 7 o3 7 or 4 7 or 3’?

De yeu scc o pottcrn5 Zbot wmhh `c ?1 tb tcrg5

o?1

7 or ?1’? 7 or ?3

Tbcrcaerc( tbc pottcrn sujjcsts tbot tbc ntb tcrg ea o J.V. ms jmvcn `y

o or n

n7 ∐?.

Tbus( o( J.V. ion `c wrmttcn os o( or ( or 0( or 9( … or n ’ ?< o( or ( or 0(...(or n ’ ?... <oiierdmnj

os J.V. ms amnmtc er mnamnmtc( rcspcitmvchy.

Tbc scrmcs o , or , or 0 , ... , or n ’? er o , or , or 0 , ... , or n ’? ,...orc iohhcd

amnmtc er mnamnmtc jcegctrmi scrmcs( rcspcitmvchy.

=.3.0. Yug te n tcrgs ea o J .V. Hct tbc amrst tcrg ea o J.V. `c o ond tbc ieggen

rotme `c r . Hct us dcnetc `y Yn tbc sug te amrst n tcrgs ea J.V. TbcnY

n7 o , or , or 0 ,..., or n ’? ... "?-

Iosc ? Ma r 7 ?( wc bovc Yn 7 o , o , o , ... , o "n tcrgs- 7 no

Iosc 0 Ma r ≦ ?( guhtmphymnj "?- `y r ( wc bovc

r Yn

7 or , or 0 , or 9 , ... , or n ... "0-

Yu`troitmnj "0- areg "?-( wc jct "? ’ r - Yn

7 o ’ or n 7 o"? ’ r n-

Tbms jmvcs"? -

Y?

n

n

o r

r

∐7

∐er

" ?-Y

?

n

n

o r

r

∐7

∐

Cxogphc = Amnd tbc ?6tb ond ntb tcrgs ea tbc J.V. 3( 03(?03(… .

Yehutmen Bcrc o 7 3 ond r 7 3. Tbus( o?6 7 3"3-?6’?

7 3"3-=

7 3?6

ond on

7 or n ’? 7 3"3-n ’? 7 3n .

Cxogphc?6 Zbmib tcrg ea tbc J.V.( 0(2(90( ... up te n tcrgs ms ?9?6:05

Yehutmen Hct ?9?6:0 `c tbc ntb tcrg ea tbc jmvcn J.V. Bcrc o 7 0 ond r 7 4.

Tbcrcaerc ?9?6:0 7 on

7 0"4-n ’ ? er 13391 7 4n ’ ?

Tbms jmvcs 42 7 4n ’ ?.

Ye tbot n ’ ? 7 2( m.c.( n 7 =. Bcnic( ?9?6:0 ms tbc =tb tcrg ea tbc J.V.



?22 GOTBCGOTMIY

Cxogphc?? Mn o J.V.( tbc 9rd tcrg ms 04 ond tbc 1tb tcrg ms ?=0.Amnd tbc ?6tb tcrg.

Yehutmen Bcrc( o or 9

0047 7 ... "?-

ond o or 1

3?=07 7 ... "0-

Dmvmdmnj "0- `y "?-( wc jct r 7 0. Yu`stmtutmnj r 7 0 mn "?-( wc jct o 7 1.

Bcnic o?6

7 1 "0-= 7 96:0.

Cxogphc?0 Amnd tbc sug ea amrst n tcrgs ond tbc sug ea amrst 3 tcrgs ea tbc jcegctrmi

scrmcs0 4

?9 =

..., , ,

Yehutmen Bcrc o 7 ? ond r 70

9. Tbcrcaerc

Yn

7

0?

9"? -

0??

9

n

no r

r

⎯ ⎤⎜ ⎞∐⎠ ⎣⎘ ⎕

⎗ ⎦∐ ⎠ ⎣⎩ ⎡7∐ ∐

70

9 ?9

n⎯ ⎤⎜ ⎞∐⎠ ⎣⎘ ⎕

⎗ ⎦⎠ ⎣⎩ ⎡

Mn portmiuhor(3

3

0Y 9 ?

9

⎯ ⎤⎜ ⎞7 ∐⎠ ⎣⎘ ⎕

⎗ ⎦⎠ ⎣⎩ ⎡7

0??9

049Ú 7

0??

2?.

Cxogphc ?9 Bew gony tcrgs ea tbc J.V.9 9

90 4

( ( (... orc nccdcd te jmvc tbc

sug961=

3?05

Yehutmen Hct n `c tbc nug`cr ea tcrgs nccdcd. Jmvcn tbot o 7 9( r 7?

0ond

961=Y

3?0n 7

Ymnic"? -

Y?

n

n

o ’ r

r 7

∐

Tbcrcaerc

?9"? -

961= ?0 1 ??3?0 0?0

n

n

∐⎜ ⎞

7 7 ∐⎘ ⎕⎗ ⎦∐



YC_SCNICY OND YCPMCY ?2=

er 961=

96:07

??

0n

∐

er ?

0n 7

961=?

96:0∐

9 ?

96:0 ?6047 7

er 0n 7 ?604 7 0?6( wbmib jmvcs n 7 ?6.

Cxogphc ?4 Tbc sug ea amrst tbrcc tcrgs ea o J.V. ms?9

?0ond tbcmr preduit ms ’ ?.

Amnd tbc ieggen rotme ond tbc tcrgs.

Yehutmen Hcto

r ( o( or `c tbc amrst tbrcc tcrgs ea tbc J.V. Tbcn

o

or or

, , 7?9

?0... "?-

ond " - " - ?o

o or ’ r

⎜ ⎞7⎘ ⎕

⎗ ⎦

... "0-

Areg "0-( wc jct o9 7 ’ ?( m.c.( o 7 ’ ? "iensmdcrmnj enhy rcoh reets-

Yu`stmtutmnj o 7 ’? mn "?-( wc bovc

? ?9?

?0 ’ ’ ’ r

r 7 er ?0r 0 , 03r , ?0 7 6.

Tbms ms o quodrotmi mn r ( sehvmnj( wc jct9 4

er 4 9

r ’ ’ 7 .

Tbus( tbc tbrcc tcrgs ea J.V. orc ;4 9 ’9 9 4 ’4

( ?( aer 7 ond ( ?( aer 79 4 4 4 9 9

’ r ’ r (

Cxogphc?3 Amnd tbc sug ea tbc scqucnic :( ::( :::( ::::( ... te n tcrgs.

Yehutmen Tbms ms net o J.V.( bewcvcr( wc ion rchotc mt te o J.V. `y wrmtmnj tbc tcrgs os

Yn 7 : , :: , ::: , :::: , ... te n tcrgs

7:

\= == === ==== te tcrgW=

... n, , , ,

70 9 4:

\"?6 ?- "?6 ?- "?6 ?- "?6 ?- tcrgsW=

...n∐ , ∐ , ∐ , ∐ ,



?=6 GOTBCGOTMIY

70 9:

\"?6 ?6 ?6 tcrgs- "?,?,?,... tcrgs-W=

...n ’ n, , ,

7: ?6"?6 ?- : ?6 "?6 ?-

= ?6 ? = =

n n

n n⎯ ⎤ ⎯ ⎤∐ ∐

∐ 7 ∐⎠ ⎣ ⎠ ⎣∐⎩ ⎡ ⎩ ⎡

.

Cxogphc ?1 O pcrsen bos 0 porcnts( 4 jrondporcnts( 2 jrcot jrondporcnts( ond se en.

Amnd tbc nug`cr ea bms onicsters durmnj tbc tcn jcncrotmens prcicdmnj bms ewn.

Yehutmen Bcrc o 7 0( r 7 0 ond n 7 ?6

Ssmnj tbc sug aerguho Yn

7" ?-

?

no r

r

∐

∐

Zc bovc Y?6

7 0"0?6 ’ ?- 7 0641

Bcnic( tbc nug`cr ea onicsters prcicdmnj tbc pcrsen ms 0641.

=.3.9 Jcegctrmi Gcon "J .G.- Tbc jcegctrmi gcon ea twe pesmtmvc nug`crs o

ond ` ms tbc nug`cr o` . Tbcrcaerc( tbc jcegctrmi gcon ea 0 ond 2 ms 4. Zc

e`scrvc tbot tbc tbrcc nug`crs 0(4(2 orc iensciutmvc tcrgs ea o J.V. Tbms hcods te o

jcncrohmsotmen ea tbc ienicpt ea jcegctrmi gcons ea twe nug`crs.

Jmvcn ony twe pesmtmvc nug`crs o ond `( wc ion mnscrt os gony nug`crs os

wc hmlc `ctwccn tbcg te golc tbc rcsuhtmnj scqucnic mn o J.V.

Hct J?( J

0(…( J

n`c n nug`crs `ctwccn pesmtmvc nug`crs o ond ` suib tbot

o(J?(J

0(J

9(…(J

n(` ms o J.V. Tbus( ` `cmnj tbc "n , 0-tb tcrg(wc bovc

?n` or ,7 (

er

?

?n`r

o

,⎜ ⎞7 ⎘ ⎕

⎗ ⎦

.

Bcnic

?

?

?Jn`

or oo

,⎜ ⎞7 7 ⎘ ⎕

⎗ ⎦(

0

?00J

nòr o

o

,⎜ ⎞7 7 ⎘ ⎕

⎗ ⎦(

9

?99J

nòr o

o

,⎜ ⎞7 7 ⎘ ⎕

⎗ ⎦(

?J

n

nnn

òr o

o

,⎜ ⎞7 7 ⎘ ⎕

⎗ ⎦

Cxogphc?: Mnscrt tbrcc nug`crs `ctwccn ? ond 031 se tbot tbc rcsuhtmnj scqucnic

ms o J.V.

Yehutmen Hct J?( J

0(J

9`c tbrcc nug`crs `ctwccn ? ond 031 suib tbot

?( J?(J

0(J

9(031 ms o J.V.



?=0 GOTBCGOTMIY

CRCPIMYC =.9

?. Amnd tbc 06tb ond ntb tcrgs ea tbc J.V.3 3 3

0 4 2 ( ( ( ...

0. Amnd tbc ?0tb tcrg ea o J.V. wbesc 2tb tcrg ms ?=0 ond tbc ieggen rotme ms 0.

9. Tbc 3tb( 2 tb ond ??tb tcrgs ea o J.V. orc p( q ond s( rcspcitmvchy. Ybew

tbot q0 7 ps.

4. Tbc 4tb tcrg ea o J.V. ms squorc ea mts sciend tcrg( ond tbc amrst tcrg ms ’ 9.

Dctcrgmnc mts :

tb

tcrg.3. Zbmib tcrg ea tbc aehhewmnj scqucnics;

"o- 0 0 0 4 ms ?02 5 ( ( (... "`- 9 9 9 9 ms:0= 5 ( ( (...

"i-? ? ? ?

ms9 = 0: ?=129 ( ( (... 5

1. Aer wbot vohucs ea x( tbc nug`crs0 0

: : ’ ( x( ’ orc mn J.V.5

Amnd tbc sug te mndmiotcd nug`cr ea tcrgs mn coib ea tbc jcegctrmi prejrcssmens mn

Cxcrimscs : te ?6;

:. 6.?3( 6.6?3( 6.66?3( ... 06 tcrgs.2. : ( 0? ( 9 : ( ... n tcrgs.

=. ?( ’ o( o0( ’ o9( ... n tcrgs "ma o ≦ ’ ?-.

?6. x9( x3( x:( ... n tcrgs "ma x ≦ ¿ ?-.

??. Cvohuotc??

?

"0 9 -l

l 7

,∟.

?0 . Tbc sug ea amrst tbrcc tcrgs ea o J.V. ms9=

?6ond tbcmr preduit ms ?. Amnd tbc

ieggen rotme ond tbc tcrgs.

?9. Bew gony tcrgs ea J.V. 9( 90

( 99

( … orc nccdcd te jmvc tbc sug ?065?4. Tbc sug ea amrst tbrcc tcrgs ea o J.V. ms ?1 ond tbc sug ea tbc ncxt tbrcc tcrgs ms

?02. Dctcrgmnc tbc amrst tcrg( tbc ieggen rotme ond tbc sug te n tcrgs ea tbc J.V.

?3. Jmvcn o J.V. wmtb o 7 :0= ond :tb tcrg 14( dctcrgmnc Y:.

?1. Amnd o J.V. aer wbmib sug ea tbc amrst twe tcrgs ms ’ 4 ond tbc amatb tcrg ms

4 tmgcs tbc tbmrd tcrg.

?:. Ma tbc 4tb( ?6 tb ond ?1tb tcrgs ea o J.V. orc x( y ond z ( rcspcitmvchy. Vrevc tbot x(

y( z orc mn J.V.



YC_SCNICY OND YCPMCY ?=9

?2. Amnd tbc sug te n tcrgs ea tbc scqucnic( 2( 22( 222( 2222… .

?=. Amnd tbc sug ea tbc preduits ea tbc ierrcspendmnj tcrgs ea tbc scqucnics 0( 4( 2(

?1( 90 ond ?02( 90( 2( 0(?

0.

06. Ybew tbot tbc preduits ea tbc ierrcspendmnj tcrgs ea tbc scqucnics o( or( or 0 (

…or n ’ ? ond O( OP( OP 0( … OP n ’ ? aerg o J.V( ond amnd tbc ieggen rotme.

0?. Amnd aeur nug`crs aergmnj o jcegctrmi prejrcssmen mn wbmib tbc tbmrd tcrg ms

jrcotcr tbon tbc amrst tcrg `y =( ond tbc sciend tcrg ms jrcotcr tbon tbc 4tb `y ?2.

00. Ma tbc ptb( qtb ond r tb tcrgs ea o J.V. orc o( ` ond i( rcspcitmvchy. Vrevc tbot

oq ’ r `r ’ pi V ’ q 7 ?.

09. Ma tbc amrst ond tbc ntb tcrg ea o J.V. orc o ond `( rcspcitmvchy( ond ma V ms tbc

preduit ea n tcrgs( prevc tbot V0 7 "o`-n.

04. Ybew tbot tbc rotme ea tbc sug ea amrst n tcrgs ea o J.V. te tbc sug ea tcrgs areg

"n , ?- tb te "0n-tb tcrg ms?n

r .

03. Ma o( `( i ond d orc mn J.V. sbew tbot

"o0 , `0 , i0- "`0 , i0 , d 0- 7 "o` , ì , id -0 .

01. Mnscrt twe nug`cr `ctwccn 9 ond 2? se tbot tbc rcsuhtmnj scqucnic ms J.V.

0:. Amnd tbc vohuc ea n se tboto `

o `

n n

n n

, ,,

,

? ?

goy `c tbc jcegctrmi gcon `ctwccn

o ond `.

02. Tbc sug ea twe nug`crs ms 1 tmgcs tbcmr jcegctrmi gcons( sbew tbot nug`crs

orc mn tbc rotme " - " -9 0 0 ; 9 0 0, ∐ .

0=. Ma O ond J `c O.G. ond J.G.( rcspcitmvchy `ctwccn twe pesmtmvc nug`crs(

prevc tbot tbc nug`crs orc O O J O J" -" -¿ , ∐ .

96. Tbc nug`cr ea òitcrmo mn o icrtomn iuhturc deu`hcs cvcry beur. Ma tbcrc wcrc 96

òitcrmo prcscnt mn tbc iuhturc ermjmnohhy( bew gony ̀ oitcrmo wmhh c̀ prcscnt ot tbc

cnd ea 0nd beur( 4tb beur ond ntb beur 5

9?. Zbot wmhh Ps 366 ogeunts te mn ?6 ycors oatcr mts dcpesmt mn o ònl wbmib poys

onnuoh mntcrcst rotc ea ?6# iegpeundcd onnuohhy5

90. Ma O.G. ond J.G. ea reets ea o quodrotmi cquotmen orc 2 ond 3( rcspcitmvchy( tbcn

e`tomn tbc quodrotmi cquotmen.



?=4 GOTBCGOTMIY

=.: Yug te nTcrgs ea Ypcimoh Ycrmcs

Zc sbohh new amnd tbc sug ea amrst n tcrgs ea segc spcimoh scrmcs( nogchy<

"m- ? , 0 , 9 ,… , n "sug ea amrst n noturoh nug`crs-

"mm- ?0 , 00 , 90 ,… , n0"sug ea squorcs ea tbc amrst n noturoh nug`crs-

"mmm- ?9 , 09 , 99 ,… , n9"sug ea iu`cs ea tbc amrst n noturoh nug`crs-.

Hct us tolc tbcg enc `y enc.

"m- Yn7? , 0 , 9 , … , n( tbcn Y

n7

" ?-

0

n n ,"Ycc Ycitmen =.4-

"mm- Bcrc Yn7 ?0 , 00 , 90 , … , n0

Zc iensmdcr tbc mdcntmty l 9 ’ "l ’ ?-9 7 9l 0 ’ 9l , ?

Vuttmnj l 7 ?( 0… suiicssmvchy( wc e`tomn

?9 ’ 69 7 9 "?-0 ’ 9 "?- , ?

09 ’ ?9 7 9 "0-0 ’ 9 "0- , ?

99 ’ 09 7 9"9-0 ’ 9 "9- , ?

.......................................

.......................................

......................................

n9 ’ "n ’ ?-9 7 9 "n-0 ’ 9 "n- , ?

Oddmnj `etb smdcs( wc jct

n9 ’ 69 7 9 "?0 , 00 , 90 , ... , n0- ’ 9 "? , 0 , 9 , ... , n- , n

9 0

? ?

9 9n n

l ’ l ’

n l ’ l n7 ,∟ ∟

@y "m-( wc lnew tbot?

" ?-? 0 9

0

n

l

n nl ... n

7

,7 , , , , 7∟

Bcnic Yn 7

0 9

?

? 9 " ?-

9 0

n

l

n n

l n n7

,⎯ ⎤

7 , ∐⎠ ⎣⎩ ⎡∟ 79 0?

"0 9 -1 n n n, ,

7" ?-"0 ?-

1

n n n, ,

"mmm- Bcrc Yn

7 ?9 , 09 , ...,n9

Zc iensmdcr tbc mdcntmty( "l , ?-4 ’ l 4 7 4l 9 , 1l 0 , 4l , ?

Vuttmnj l 7 ?( 0( 9… n( wc jct



YC_SCNICY OND YCPMCY ?=3

04 ’ ?4 7 4"?-9 , 1"?-0 , 4"?- , ?

94 ’ 04 7 4"0-9 , 1"0-0 , 4"0- , ?

44 ’ 94 7 4"9-9 , 1"9-0 , 4"9- , ?

..................................................

..................................................

..................................................

"n ’ ?-4 ’ "n ’ 0-4 7 4"n ’ 0-9 , 1"n ’ 0-0 , 4"n ’ 0- , ?

n4

’ "n ’ ?-4

7 4"n ’ ?-9

, 1"n ’ ?-0

, 4"n ’ ?- , ?"n , ?-4 ’ n4 7 4n9 , 1n0 , 4n , ?

Oddmnj `etb smdcs( wc jct

"n , ?-4 ’ ?4 7 4"?9 , 09 , 99 ,..., n9- , 1"?0 , 00 , 90 , ..., n0- ,

4"? , 0 , 9 ,..., n- , n

9 0

? ? ?

4 1 4n n n

l l l

l l l n7 7 7

7 , , ,∟ ∟ ∟ ... "?-

Areg ports "m- ond "mm-( wc lnew tbot

0

? ?

" ?- " ?- "0 ?-ond0 1

n n

l l

n n n n nl l 7 7

, , ,7 7∟ ∟Vuttmnj tbcsc vohucs mn cquotmen "?-( wc e`tomn

9 4 9 0

?

1 " ?- "0 ?- 4 " ?-4 4 1 4

1 0

n

l

n n n n nl n n n n ’ ’ ’ n

7

, , ,7 , , ,∟

er 4Yn7 n4 , 4n9 , 1n0 , 4n ’ n "0n0 , 9n , ?- ’ 0n "n , ?- ’ n

7 n4 , 0n9 , n0

7 n0"n , ?-0.

Bcnic( Yn 7 \ W

00 0 " ?-" ?-

4 4

n nn n ,,7

Cxogphc ?= Amnd tbc sug te n tcrgs ea tbc scrmcs; 3 , ?? , ?= , 0= , 4?…

Yehutmen Hct us wrmtc

Yn7 3 , ?? , ?= , 0= , ... , o

n ’?, o

n

er Yn7 3 , ?? , ?= , ... , o

n ’0, o

n’ ?, o

n

En su`troitmen( wc jct



?=1 GOTBCGOTMIY

6 7 3 , \1 , 2 , ?6 , ?0 , ..."n ’ ?- tcrgsW ’ on

er on

7 3 ," ?-\?0 " 0- 0W

0

n ’ n, ∐ Ú

7 3 , "n ’ ?- "n , 4- 7 n0 , 9n , ?

Bcnic Yn

70 0

? ? ? ?

" 9 ?- 9n n n n

l

l l l

o l l l l n7 7 7

7 , , 7 , ,∟ ∟ ∟ ∟

7" ?-"0 ?- 9 " ?-

1 0

n n n n nn, , ,, ,

" 0-" 4-

9

n n n, ,7 .

Cxogphc 06 Amnd tbc sug te n tcrgs ea tbc scrmcs wbesc nt b tcrg ms n "n,9-.

Yehutmen Jmvcn tbot on

7 n "n , 9- 7 n0 , 9n

Tbus( tbc sug te n tcrgs ms jmvcn `y

Yn

70

? ? ?

9n n n

l

l l l

o l l 7 7 7

7 ,∟ ∟ ∟

7 " ?- "0 ?- 9 " ?-1 0

n n n n n, , ,, " ?-" 3-9

n n n, ,7 .

CRCPIMYC =.4

Amnd tbc sug te n tcrgs ea coib ea tbc scrmcs mn Cxcrimscs ? te :.

?. ? Ú 0 , 0 Ú 9 , 9 Ú 4 , 4 Ú 3 ,... 0. ? Ú 0 Ú 9 , 0 Ú 9 Ú 4 , 9 Ú 4 Ú 3 , ...

9. 9 Ú ?0 , 3 Ú 00 , : Ú 90 , ... 4.? ? ?

? 0 0 9 9 4, , ,

Ú Ú Ú ...

3. 30 , 10 , :0 , ... , 060 1. 9 Ú 2 , 1 Ú ?? , = Ú ?4 , ...

:. ?0 , "?0 , 00- , "?0 , 00 , 90- , ...

Amnd tbc sug te n tcrgs ea tbc scrmcs mn Cxcrimscs 2 te ?6 wbesc ntb tcrgs ms jmvcn `y

2. n "n,?- "n,4-. =. n0 , 0n

?6. 0"0 ?-n ’



YC_SCNICY OND YCPMCY ?=:

Gmsichhonceus Cxogphcs

Cxogphc0? Ma ptb( qtb( r tb ond stb tcrgs ea on O.V. orc mn J.V( tbcn sbew tbot

" p ’ q-( "q ’ r -( "r ’ s- orc ohse mn J.V.

Yehutmen Bcrc

o p

7 o , " p ’?- d ... "?-

oq

7 o , "q ’?- d ... "0-

or

7 o , "r ’?- d ... "9-

o s

7 o , " s ’?- d ... "4-

Jmvcn tbot o p( o

q( o

r ond o

s orc mn

J.V

.(

Ye q q r r

p q p q

o o oo q r

o o o o p q

∐ ∐7 7 7

∐ ∐ "wby 5-... "3-

Ymgmhorhy s r sr

q r q r

o o oo r s

o o o o q r

∐ ∐7 7 7

∐ ∐ "wby 5-... "1-

Bcnic( `y "3- ond "1-

q ’ r r ’ s p ’ q q ’ r

7 ( m.c.( p ’ q( q ’ r ond r ’ s orc mn J.V.

Cxogphc 00 Ma o( `( i orc mn J.V. ond?? ?

y x z o ` i7 7 ( prevc tbot x( y( z orc mn O.V.

Yehutmen Hct

?

xo 7

? ?

y z ` i l 7 7 Tbcn

o 7 l x ( ` 7 l y ond i 7 l z . ... "?-

Ymnic o( `( i orc mn J.V.( tbcrcaerc(

`0 7 oi ... "0-

Ssmnj "?- mn "0-( wc jct

l 0 y

7 l x , z

( wbmib jmvcs 0 y 7 x , z.Bcnic( x( y ond z orc mn O.V.

Cxogphc 09 Ma o( `( i( d ond p orc dmaacrcnt rcoh nug`crs suib tbot

"o0 , `0 , i0- p0 ’ 0"o` , `i , id - p , "`0 , i0 , d 0- ≤ 6( tbcn sbew tbot o( `( i ond d

orc mn J.V.

Yehutmen Jmvcn tbot

"o0 , `0 , i0- p0 ’ 0 "o` , `i , id - p , "`0 , i0 , d 0- ≤ 6 ... "?-



?=2 GOTBCGOTMIY

@ut H.B.Y.

7 "o0 p0 ’ 0o`p , `0- , "`0 p0 ’ 0`ip , i0- , "i0 p0 ’ 0idp , d 0-(

wbmib jmvcs "op ’ `-0 , "`p ’ i-0 , "ip ’ d -0 ≣ 6 ... "0-

Ymnic tbc sug ea squorcs ea rcoh nug`crs ms nen ncjotmvc( tbcrcaerc( areg "?- ond "0-(

wc bovc( "op ’ `-0 , "`p ’ i-0 , "ip ’ d -0 7 6

er op ’ ` 7 6 ( `p ’ i 7 6 ( ip ’ d 7 6

Tbms mgphmcs tbot ` i d o ` i

7 7 7

Bcnic o( `( i ond d orc mn J.V.

Cxogphc 04 Ma p(q(r orc mn J.V. ond tbc cquotmens( px0 , 0qx , r 7 6 ond

dx0 , 0cx , a 7 6 bovc o ieggen reet( tbcn sbew tbotd

p

c

q

a

r ( ( orc mn O.V.

Yehutmen Tbc cquotmen px0 , 0qx , r 7 6 bos reets jmvcn `y

00 4 4

0

q q rp x

p

∐ ¿ ∐7

Ymnic p (q( r orc mn J.V. q0 7 pr . Tbusq

x p

∐7 `ut

q

p

∐ms ohse reet ea

dx0 , 0cx , a 7 6 "Zby 5-. Tbcrcaerc

0

0 6q q

d c a ( p p

⎜ ⎞ ⎜ ⎞∐ ∐, , 7⎘ ⎕ ⎘ ⎕

⎗ ⎦ ⎗ ⎦

er dq0 ’ 0cqp , ap0 7 6 ... "?-

Dmvmdmnj "?- `y pq0

ond usmnj

q0

7 pr ( wc jct

06

d c ap (

p q pr ∐ , 7

er

0c d a

q p r 7 ,

Bcnicd c a

( ( p q r

orc mn O.V.



YC_SCNICY OND YCPMCY ?==

Gmsichhonceus Cxcrimsc En Iboptcr =

?. Ybew tbot tbc sug ea "g , n-tb ond "g ’ n-tb tcrgs ea on O.V. ms cquoh te twmic

tbc gtb tcrg.

0. Ma tbc sug ea tbrcc nug`crs mn O.V.( ms 04 ond tbcmr preduit ms 446( amnd tbc

nug`crs.

9. Hct tbc sug ea n( 0n( 9n tcrgs ea on O.V. `c Y?( Y

0ond Y

9( rcspcitmvchy( sbew tbot

Y9

7 9"Y0

’ Y?-

4. Amnd tbc sug ea ohh nug`crs `ctwccn 066 ond 466 wbmib orc dmvmsm`hc `y :.

3. Amnd tbc sug ea mntcjcrs areg ? te ?66 tbot orc dmvmsm`hc `y 0 er 3.1. Amnd tbc sug ea ohh twe dmjmt nug`crs wbmib wbcn dmvmdcd ̀ y 4( ymchds ? os

rcgomndcr.

:. Ma a ms o aunitmen sotmsaymnj a " x ,y- 7 a " x- a " y- aer ohh x( y ∂ N suib tbot

a "?- 7 9 ond?

" - ?06n

x

a x7

7∟ ( amnd tbc vohuc ea n.

2. Tbc sug ea segc tcrgs ea J.V. ms 9?3 wbesc amrst tcrg ond tbc ieggen rotme orc

3 ond 0( rcspcitmvchy. Amnd tbc host tcrg ond tbc nug`cr ea tcrgs.

=. Tbc amrst tcrg ea o J.V. ms ?. Tbc sug ea tbc tbmrd tcrg ond amatb tcrg ms =6.

Amnd tbc ieggen rotme ea J.V.

?6. Tbc sug ea tbrcc nug`crs mn J.V. ms 31. Ma wc su`troit ?( :( 0? areg tbcsc nug`crsmn tbot erdcr( wc e`tomn on ormtbgctmi prejrcssmen. Amnd tbc nug`crs.

??. O J.V. iensmsts ea on cvcn nug`cr ea tcrgs. Ma tbc sug ea ohh tbc tcrgs ms 3 tmgcs

tbc sug ea tcrgs eiiupymnj edd phoics( tbcn amnd mts ieggen rotme.

?0. Tbc sug ea tbc amrst aeur tcrgs ea on O.V. ms 31. Tbc sug ea tbc host aeur tcrgs ms

??0. Ma mts amrst tcrg ms ??( tbcn amnd tbc nug`cr ea tcrgs.

?9. Ma o `x

o `x

` ix

` ix

i dx

i dx x

,

∐7

,

∐7

,

∐≦" - (6 tbcn sbew tbot o( `( i ond d orc mn J.V.

?4. Hct Y `c tbc sug( V tbc preduit ond P tbc sug ea rcimpreiohs ea n tcrgs mn o J.V.

Vrevc tbot V0P n 7 Yn.

?3. Tbc ptb

( qtb

ond r tb

tcrgs ea on O.V. orc o( `( i( rcspcitmvchy. Ybew tbot"q ’ r -o , "r ’ p -` , " p ’ q -i 7 6

?1. Ma ? ? ? ? ? ?

o (` (i` i i o o `

⎜ ⎞ ⎜ ⎞ ⎜ ⎞, , ,⎘ ⎕ ⎘ ⎕ ⎘ ⎕

⎗ ⎦ ⎗ ⎦ ⎗ ⎦orc mn O.V.( prevc tbot o( `( i orc mn O.V.

?:. Ma o( `( i( d orc mn J.V( prevc tbot "on , `n-( "`n , in-( "in , d n- orc mn J.V.

?2 . Ma o ond ` orc tbc reets ea x0 ’ 9 x , p 7 6 ond i( d orc reets ea x0 ’ ?0 x , q 7 6(

wbcrc o( `( i( d aerg o J.V. Vrevc tbot "q , p- ; "q ’ p- 7 ?:;?3.



066 GOTBCGOTMIY

?=. Tbc rotme ea tbc O.G. ond J.G. ea twe pesmtmvc nug`crs o ond `( ms g ; n. Ybew

tbot " - " -0 0 0 0;o ` g g ’ n ; g ’ g ’ n7 , .

06. Ma o( `( i orc mn O.V.< `( i( d orc mn J.V. ond? ? ? ( (

i d corc mn O.V. prevc tbot o( i( c

orc mn J.V.

0?. Amnd tbc sug ea tbc aehhewmnj scrmcs up te n tcrgs;

"m- 3 , 33 ,333 , … "mm- .1 ,. 11 ,. 111,…

00. Amnd tbc 06tb tcrg ea tbc scrmcs 0 Ú 4 , 4 Ú 1 , 1 Ú 2 , ... , n tcrgs.

09. Amnd tbc sug ea tbc amrst n tcrgs ea tbc scrmcs; 9, : ,?9 ,0? ,9? ,…

04. Ma Y?( Y

0( Y

9orc tbc sug ea amrst n noturoh nug`crs( tbcmr squorcs ond tbcmr

iu`cs( rcspcitmvchy( sbew tbot = 0

0Y 7 Y

9"? , 2Y

?-.

03. Amnd tbc sug ea tbc aehhewmnj scrmcs up te n tcrgs;

9 9 0 9 9 9? ? 0 ? 0 9

? ? 9 ? 9 3...

, , ,, , ,

, , ,

01. Ybew tbot0 0 0

0 0 0

? 0 0 9 " ?- 9 3

9 ?? 0 0 9 " ?-

... n n n

n... n n

Ú , Ú , , Ú , ,7

,Ú , Ú , , Ú , .0:. O aorgcr ùys o uscd troiter aer Ps ?0666. Bc poys Ps 1666 iosb ond ojrccs te

poy tbc ̀ ohonic mn onnuoh mnstohgcnts ea Ps 366 phus ?0# mntcrcst en tbc unpomd

ogeunt. Bew guib wmhh tbc troiter iest bmg5

02. Ybogsbod Ohm ùys o sieetcr aer Ps 00666. Bc poys Ps 4666 iosb ond ojrccs te

poy tbc ̀ ohonic mn onnuoh mnstohgcnt ea Ps ?666 phus ?6# mntcrcst en tbc unpomd

ogeunt. Bew guib wmhh tbc sieetcr iest bmg5

0=. O pcrsen wrmtcs o hcttcr te aeur ea bms armcnds. Bc osls coib enc ea tbcg te iepy

tbc hcttcr ond gomh te aeur dmaacrcnt pcrsens wmtb mnstruitmen tbot tbcy gevc tbc

ibomn smgmhorhy. Ossugmnj tbot tbc ibomn ms net ̀ relcn ond tbot mt iests 36 pomsc te

gomh enc hcttcr. Amnd tbc ogeunt spcnt en tbc pestojc wbcn 2 tb sct ea hcttcr ms

gomhcd.96. O gon dcpesmtcd Ps ?6666 mn o ònl ot tbc rotc ea 3# smgphc mntcrcst onnuohhy.

Amnd tbc ogeunt mn ?3tb ycor smnic bc dcpesmtcd tbc ogeunt ond ohse iohiuhotc tbc

tetoh ogeunt oatcr 06 ycors.

9?. O gonuaoiturcr rcilens tbot tbc vohuc ea o goibmnc( wbmib iests bmg Ps. ?3103(

wmhh dcprcimotc coib ycor ̀ y 06#. Amnd tbc cstmgotcd vohuc ot tbc cnd ea 3 ycors.

90. ?36 werlcrs wcrc cnjojcd te amnmsb o ke` mn o icrtomn nug`cr ea doys. 4 werlcrs

dreppcd eut en sciend doy( 4 gerc werlcrs dreppcd eut en tbmrd doy ond se en.



YC_SCNICY OND YCPMCY 06?

Mt teel 2 gerc doys te amnmsb tbc werl. Amnd tbc nug`cr ea doys mn wbmib tbc werl

wos iegphctcd.

Yuggory

@y o scqucnic( wc gcon on orronjcgcnt ea o nug`cr mn o dcamnmtc erdcr

oiierdmnj te segc ruhc. Ohse( wc dcamnc o scqucnic os o aunitmen wbesc degomn

ms tbc sct ea noturoh nug`crs er segc su`scts ea tbc typc {?( 0( 9....l -. O

scqucnic ientomnmnj o amnmtc nug`cr ea tcrgs ms iohhcd o amnmtc scqucnic. Oscqucnic ms iohhcd mnamnmtc ma mt ms net o amnmtc scqucnic.

Hct o? ( o

0 ( o

9( ... `c tbc scqucnic( tbcn tbc sug cxprcsscd os o

? , o

0 , o

9 , ...

ms iohhcd scrmcs. O scrmcs ms iohhcd amnmtc scrmcs ma mt bos jet amnmtc nug`cr ea

tcrgs.

On ormtbgctmi prejrcssmen "O.V.- ms o scqucnic mn wbmib tcrgs mnircosc er

dcircosc rcjuhorhy `y tbc sogc ienstont. Tbms ienstont ms iohhcd ieggen

dmaacrcnic ea tbc O.V. Ssuohhy( wc dcnetc tbc amrst tcrgs ea O.V. `y o( tbc

ieggen dmaacrcnic `y d ond tbc host tcrg `y h . Tbc jcncroh tcrg er tbc ntb

tcrg ea tbc O.V. ms jmvcn `y on

7 o , "n ’ ?- d .

Tbc sug Yn

ea tbc amrst n tcrgs ea on O.V. ms jmvcn `y

" - " -Y ?n

n n7 0o , n ’ d 7 o ,h

0 0⎯ ⎤⎩ ⎡ .

Tbc ormtbgctmi gcon O ea ony twe nug`crs o ond ` ms jmvcn `y0

o ,`m.c.( tbc

scqucnic o( O( ` ms mn O.V.

O scqucnic ms somd te `c o jcegctrmi prejrcssmen er J.V.( ma tbc rotme ea ony

tcrg te mts prcicdmnj tcrg ms sogc tbreujbeut. Tbms ienstont aoiter ms iohhcd

tbc ieggen rotme. Ssuohhy( wc dcnetc tbc amrst tcrg ea o J.V. `y o ond mts

ieggen rotme `y r . Tbc jcncroh er tbc ntb tcrg ea J.V. ms jmvcn `y on7 or n ’ ?.

Tbc sug Yn

ea tbc amrst n tcrgs ea J.V. ms jmvcn `y



060 GOTBCGOTMIY

" - " - ’ ? ?’ Y ?

? ? ’

n n

n

o r o r 7 er ( ma r

r ’ r ≦

Tbc jcegctrmi gcon "J.G.- ea ony twe pesmtmvc nug`crs o ond ` ms jmvcn `y

o` m.c.( tbc scqucnic o( J( ` ms J.V.

Bmstermioh Netc

Cvmdcnic ms aeund tbot @o`yhenmons( segc 4666 ycors oje( lncw ea ormtbgctmi ond

jcegctrmi scqucnics. Oiierdmnj te @ectbmus "3?6 O.D.-( ormtbgctmi ond jcegctrmi

scqucnics wcrc lnewn te corhy Jrccl wrmtcrs. Ogenj tbc Mndmon gotbcgotmimon(

Oryo`botto "4:1 O.D.- wos tbc amrst te jmvc tbc aerguho aer tbc sug ea squorcs ond

iu`cs ea noturoh nug`crs mn bms aogeus werl Oryo`botmyog( wrmttcn oreund

4== O.D. Bc ohse jovc tbc aerguho aer amndmnj tbc sug te n tcrgs ea on ormtbgctmi

scqucnic stortmnj wmtb ptb tcrg. Netcd Mndmon gotbcgotmimons @robgjupto

"3=2 O.D.-( Gobovmro "236 O.D.- ond @bosloro "???4'??23 O.D.- ohse iensmdcrcd

tbc sug ea squorcs ond iu`cs. Onetbcr spcimami typc ea scqucnic bovmnj mgpertontopphmiotmens mn gotbcgotmis( iohhcd Am`enoiim scqucnic( wos dmsievcrcd `y Mtohmon

gotbcgotmimon Hcenorde Am`enoiim "??:6'?036 O.D.-. Ycvcntccntb icntury wmtncsscd

tbc ihossmamiotmen ea scrmcs mnte spcimami aergs. Mn ?1:? O.D. Kogcs Jrcjery uscd

tbc tcrg mnamnmtc scrmcs mn ienncitmen wmtb mnamnmtc scqucnic. Mt wos enhy tbreujb tbc

rmjereus dcvchepgcnt ea ohjc`romi ond sct tbcerctmi teehs tbot tbc ienicpts rchotcd

te scqucnic ond scrmcs ieuhd `c aerguhotcd sumto`hy.

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class 11th maths chapter 9

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