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>

YC^SCFICY NFB YCPECYEfrplbsirelf=N yc~scfic ey n ycr lg rcpky ef n bcgeferc lpbcp tero n psac lg l`rnefefd ery rcpky& Acr F ;z>! 0! 9! { `c n ycr lg xlyerewc ilfycisrewc efrcdcpy&

Ef dcfcpna! n yc~scfic knu `c bcgefcb ny n knxxefd g = F _! tocpc _ ey n flf ckxru

ycr& Roc cackcfry lg roc pnfdc ycr _ npc bcflrcb `u n >! n0! !nf! ! ef toeio nf ey rocdcfcpna rcpk lg roc yc~scfic& Tc tperc roc yc~scfic ny zn f{ lp zn>! n0! n9! ! nf! {&

Rocpc ey n lfc rl lfc ilppcyxlfbcfic `crtccf roc cackcfry lg roc blknef ycr F nfbroc pnfdc ycr _& Eg roc blknef ycr F ony n geferc fsk`cp lg cackcfry! rocf roc pnfdc ycr_ ony nayl n geferc fsk`cp lg cackcfry nfb roc yc~scfic zn >! n0! ! nf{ ey inaacb n gefercyc~scfic& Eg roc blknef ycr F ony efgeferc fsk`cp lg cackcfry! rocf roc pnfdc ycr naylony nf efgeferc fsk`cp lg cackcfry nfb tc inaa roc yc~scfic nf efgeferc yc~scfic&

Roc ysk n> * n0 * n9* * nf* ! lg roc cackcfry lg roc yc~scfic znf{ eyinaacb n ycpecy& n> ey inaacb roc gepyr rcpk lg roc ycpecy! n0 ey inaacb roc ycilfb rcpk lg rocycpecy nfb nf ey roc fro rcpk lp roc dcfcpna rcpk lg roc ycpecy& Roc ycpecy ey geferc eg rocyc~scfic znf{ ey geferc nfb roc ycpecy ey efgeferc eg roc yc~scfic ey efgeferc& Eg roc rcpk lg n

yc~scfic glaalt n icprnef xnrrcpf! rocf roc yc~scfic ey inaacb n xpldpcyyelf&

Nperokcrei Xpldpcyyelf /N&X&%=

N yc~scfic lg fsk`cpy ey yneb rl `c ef N&X& tocf roc beggcpcfic `crtccf nfu rtlilfycisrewc fsk`cpy ey natnuy ynkc& Eg n ey roc gepyr rcpk nfb b roc ilkklf beggcpcfic!roc N&X& inf `c tperrcf ny n! n * b! n * 0b!&! n * /f >%b& Roc fro rcpk rf ey dewcf `u rf; n * /f >%b& Roc ysk Yf lg roc gepyr f rcpky lg ysio nf N&X& ey dewcf `u

ff f

Y 0n /f >%b n0 0

a

tocpc a ey roc anyr rcpk /e&c& roc fro rcpk lg roc N&X&%&Flrcy=

Eg n gecb fsk`cp ey nbbcb /ys`rpnircb% rl cnio rcpk lg n dewcf N&X& rocf rocpcysarefd yc~scfic ey nayl nf N&X& tero roc ynkc ilkklf beggcpcfic ny ronr lg

roc dewcf N&X&

Eg cnio rcpk lg nf N&X& ey ksarexaecb `u n gecb fsk`cp/ynu j% /lp bewebcb `u n

flf+}cpl gecb fsk`cp%! roc pcysarefd yc~scfic ey nayl nf N&X& tero roc

ilkklf beggcpcfic ksarexaecb `u j&

Eg n>! n0! n9!&nfb `>! `0! `9!&npc rtl N&X&y tero ilkklf beggcpcficy b nfb bpcyxcirewcau rocf n> * `>! n0 * `0! n9 * `9! ey nayl nf N&X& tero ilkklf

beggcpcfic b * b & Eg nf nfb `fnpc roc fro rcpky lg roc rtl N&X&y! rocf roc fro

rcpk lg roc pcysarefd N&X& ey nf * `f ; n * /f >%b * ` * /f >%b ; /n * `% * /f

>% /b * b %& Eg tc onwc rl rnjc ropcc rcpky ef nf N&X&! er ey ilfwcfecfr rl rnjc rock ny n b!

n!n * b& Ef dcfcpna! tc rnjc n pb! n /p>%b! &! n b! n! n * b! &! n * pb! ef

inyc tc onwc rl rnjc /0p * >% rcpky ef nf N&X&

Eg tc onwc rl rnjc glsp rcpky! tc rnjc n 9b! n b! n * b! n * 9b& Ef dcfcpna!

tc rnjc n/0p >%b! n /0p 9%b!&&&&! nb! n * b! &! n * /0p >%b! ef inyc tc

onwc rl rnjc 0p rcpky ef nf N&X&

8/4/2019 Wikimama Class 11 Ch 9 Sequence and Series

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Eg n>! n0! n9! ! nf npc ef N&X&! rocf n> * nf ; n0 * nf+> ; n9 * nf 0; & nfb yl

lf&

Nperokcrei Kcnf/y%= Eg ropcc rcpky npc ef N&X&! rocf roc kebbac rcpk ey inaacb roc nperokcrei kcnf

/N&K&% `crtccf roc lrocp rtl e&c& eg n!`!i npc ef N&X& rocf n i`0

ey roc N&K& lg

n nfb i& Eg n>! n0!&! nf npc f fsk`cpy rocf roc nperokcrei kcnf /N% lg rocyc fsk`cpy ey

> 0 9 f>

N /n n n &&&&&& n %&f

Roc f fsk`cpy N>! N0! & ! Nfnpc yneb rl `c N&K&y `crtccf roc fsk`cpy nnfb ` eg n! N>! N0! &! Nf! ` npc ef N&X& Eg b ey roc ilkklf beggcpcfic lg roey N&X&

rocf ` ; n * /f * 0 >%b` n

bf >

> 0 f` n fn ` 0/` n% f/` n% n f`N n ! N n !&&&&&&!N nf > f > f > f > f >

&

Dclkcrpei Xpldpcyyelf /D&X&%

N D&X& ey n yc~scfic tolyc gepyr rcpk ey flf+}cpl nfb cnio lg tolyc ysiiccbefd rcpkey p rekcy roc xpcicbefd rcpk! tocpc p ey ylkc gecb flf+}cpl fsk`cp! jfltf ny rocilkklf pnrel lg roc D&X& Eg n ey roc gepyr rcpk lg n D&X& nfb p ery ilkklf pnrel! rocf eryfro rcpk rf ey dewcf `u rf ; np

f+>& Roc ysk Yf lg roc gepyr f rcpky lg roc D&X& eyf

fn/p >%

Y ! p >p >

fn! p >&Flrcy= Eg cnio rcpk lg n D&X& ey ksarexaecb /bewebcb% `u n gecb flf+}cpl ilfyrnfr! rocf

roc pcysarefd yc~scfic ey nayl n D&X& tero ynkc pnrel ny ronr lg roc dewcf D&X& & Eg cnio rcpk lg n D&X& /tero ilkklf pnrel p% ey pneycb rl roc xltcp j! rocf roc

pcysarefd yc~scfic ey nayl n D&X& tero ilkklf pnrel pj&Eg roc dewcf D&X& ey n! np! np0! nfb cnio rcpk ey pneycb rl roc jro xltcp!

tc l`rnef roc yc~scfic nj! njpj! njp0j! & toeio ey nayl n D&X& tero nj nyroc gepyr rcpk nfb pj ny roc ilkklf pnrel&

Eg n>! n0! n9!& nfb `>! `0! `9!& npc rtl D&X&y tero ilkklf pnrely p nfb p

pcyxcirewcau rocf roc yc~scfic n>`>! n0`0! n9`9! & ey nayl n D&X& tero ilkklf

pnrel pp & Eg tc onwc rl rnjc ropcc rcpky ef n D&X&! er ey ilfwcfecfr rl rnjc rock ny n,p! n!

np&

Ef dcfcpna! tc rnjc jj j >

n n! !&&&&! n! np !&&&! np

p pef inyc tc onwc rl rnjc /0j

* >% rcpky ef n D&X&

8/4/2019 Wikimama Class 11 Ch 9 Sequence and Series

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Eg tc onwc rl rnjc glsp rcpky ef n D&X&! er ey ilfwcfecfr rl rnjc rock ny n,p 9! n,p!

np! np9& Ef dcfcpna! tc rnjc 0j >0j > 0j 9

n n n! !&&&&! ! np!&&&&! np

pp p! ef inyc tc

onwc rl rnjc 0j rcpky ef n D&X& Eg n>! n0!& !nf npc ef D&X&! rocf n>nf ; n0 nf+> ; n9 nf+0; & Eg n>! n0! n9!& ey n D&X& /cnio nE 6 8%! rocf aldn>! aldn0! aldn9 & ey nf N&X& Roc

ilfwcpyc ey nayl rpsc&

Dclkcrpei Kcnf/y%= Eg ropcc rcpky npc ef D&X&! rocf roc kebbac rcpk ey inaacb roc dclkcrpei kcnf

/D&K&% `crtccf roc rtl& Yl eg n! `! i npc ef D&X& rocf ` ; ni ey roc dclkcrpei

kcnf lg n nfb i& Eg n>! n0!&! nf npc xlyerewc fsk`cpy rocf rocep D&K&/D% ey dewcf `u D ;

/n>n0n9&nf%>,f&

Eg D>! D0!&Df npc f dclkcrpei kcnfy `crtccf n nfb ` rocf n! D>! D0!&! Df!` teaa `c n D&X& Ocpc ` ; npf * >

p ; f >`

nD> ; n f >

`

n! D0 ; n

0

f >`

n!& ! Df ; n

f

f >`

n&

Nperokcreil+Dclkcrpei Xpldpcyyelf

Ysxxlyc n>! n0! n9!& ey nf N&X& nfb `>! `0! `9!& ey n D&X& Rocf roc yc~scfic n>`>!

n0`0!&! nf`f ey yneb rl `c nf nperokcreil+dclkcrpei xpldpcyyelf& Nf nperokcreil+

dclkcrpei xpldpcyyelf ey lg roc glpk n`! /n * b%`p! /n * 0b%`p 0! /n * 9b%`p9!&

Ery ysk Yf

rl f rcpky ey dewcf `u

Yf ; n` * /n * b%`p * /n * 0b%`p0*&*/n * /f 0%b%`pf+0 * /n * /f + >%b%`pf+>&

Ksarexau `lro yebcy `u p! yl ronr

pYf ; n`p * /n * b%`p0*& */n * /f 9%b%`pf 0 * /n * /f 0%b%`pf > * /n * /f

>%b%`pf&

Ys`rpnirefd tc dcr

/> p%Yf ; n` * b`p * b`p0 *&&&&&&&* b`pf 0 * b`pf > /n * /f >%b%`pf

;f >

fb`p/> p %n` n f > b p/> p%

ff >

f 0

n /f >%b `pn` b`p/> p %Y > p > p/> p%

&

Eg > 1 p 1 >! roc ysk lg roc efgeferc fsk`cp lg rcpky lg roc xpldpcyyelf ey

f 0f

n` b`paek Y

> p /> p %&

/> %+> ; > * * 0 * 9*& > 1 1 >&

/> %+0 ; > * 0 * 90*& > 1 1 >&

8/4/2019 Wikimama Class 11 Ch 9 Sequence and Series

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Onpklfei Xpldpcyyelf /O&X&%

Roc yc~scfic n>! n0! n9!&! nf!& /ne 8% ey yneb rl `c ef O&X& eg roc yc~scfic

> 0 9 f

> > > >

! ! !&&&&&& !&&&&&&n n n n ey ef N&X& Roc f

ro

rcpk lg roc O&X& ey f

>

n n /f >%b ! tocpc

>

>n

n 0 >

> >nfb b &

n n

Onpklfei Kcnf/y%=

Eg n nfb ` npc rtl flf+}cpl fsk`cpy! rocf roc onpklfei kcnf lg n nfb ` ey nfsk`cp O ysio ronr roc fsk`cpy n! O! ` npc ef O&X& Tc onwc> > > > 0n`

/ % O &O 0 n ` n `

Eg n>! n0!&!nf npc f flf+}cpl fsk`cpy! rocf roc onpklfei kcnf O lg rocyc

fsk`cp ey dewcf `u> 0 f

> > > > >&&&&&&&

O f n n nO ;

> 0 f

f

> > >

n n n

&

Roc f fsk`cpy O>! O0!&!Of npc yneb rl `c onpklfei kcnfy `crtccf n nfb `!eg

n! O>! O0!& !Of! ` npc ef O&X& e&c eg> 0 f

> > > > >! ! ! &&&&&! !

n O O O `npc ef N&X& Acr

b `c roc ilkklf beggcpcfic lg roc ilppcyxlfbefd N&X&! Rocf> >

f > b` n

b ;n `

f > n`&

Rosy> 0 f

0 n f n > > n n f` > > > > fn ! !&&&&!

O n /f >%n` /f >%n` O n f > n` O n f > n` /f >%n`&

Kcrolb lg Beggcpcficy

Ysxxlyc n>! n0! n9!& ey n yc~scfic ysio ronr roc yc~scfic n0 n>! n9 n0!& ey cerocpnf N&X& lp n D&X&& Roc f ro rcpk! nf! lg roey yc~scfic ey l`rnefcb ny glaalty=Y ; n> * n0 * n9*& *nf+> * nf & />%

Y ; n> * n0*& *nf+0 * nf+> * nf & /0%Ys`rpnirefd /0% gplk />%! tc dcr! nf ; n> * Z/n0 n>% * /n9 n0% *& * /nf nf + >%[&Yefic roc rcpky teroef roc `pnijcry npc cerocp ef nf N&X& lp n D&X&! tc inf gefb roc wnasclg nf! roc f

ro rcpk& Tc inf flt gefb roc ysk lg roc f rcpky lg roc yc~scfic nyf

jj >

Y n &

8/4/2019 Wikimama Class 11 Ch 9 Sequence and Series

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3

Eg ilppcyxlfbefd rl roc yc~scfic n>! n0! n9!&!nf! rocpc ceyry n yc~scfic `8! `>! `0!&!`f ysio ronr nj ; `j `j+>! /j ; >! 0! 9! &% rocf roc ysk lg f rcpky lg roc yc~scfic n>!n0!nf ey `f `8&

Ylkc Ekxlprnfr Pcysary

> * 0 * 9 ** f ; f0

/f * >% /ysk lg gepyr f fnrspna fsk`cpy%&

Efc~snaerecy

N&K& D& K& O& K& =

Glp rtl xlyerewc fsk`cpy n nfb `!

N&K& ;n `

0! D& K& ; n` ! O&K& ;

0n`

n `yl ronr

N&K& D&K& ;

0

n `n ` 0 n` 80 0

nfb D&K& O&K& ;n` n ` 0 n`0n`

n`n n

;

0n` n `

8n `

N&K D&K& O&K&

Acr n>! n0!&! nf `c f xlyerewc pcna fsk`cpy! rocf tc bcgefc rocep nperokcrei kcnf

/N%! dclkcrpei kcnf /D% nfb onpklfei kcnf /O% ny N ;> 0 fn n &&&& n

f ! D ; /n> n0

nf%>,f nfb O ;

> 0 9 f

f

> > > >&&&&

n n n n

&

Er inf `c yoltf ronr N D O& Klpclwcp c~snaeru olaby nr cerocp xanic eg nfb

lfau eg

n> ; n0;& ; nf&

Tcedorcb Kcnfy=

Acr n>! n0!& ! nf `c f xlyerewc pcna fsk`cpy nfb k>! k0!&! kf `c f xlyerewc pnrelfnafsk`cpy& Rocf tc bcgefc tcedorcb Nperokcrei kcnf /N-%! tcedorcb Dclkcrpei kcnf/D-% nfb tcedorcb onpklfei kcnf /O-% ny

N- ; > > 0 0 f f

> 0 f

k n k n &&&& k n

k k &&&& k! D-; > 0 f > 0 f

>k k k k k &&&&&& k

> 0 fn &n & &&&& n nfb

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O- ; > 0 f

> 0 f

> 0 f

k k &&&& k

k k k&&&&

n n n

&

Er inf `c yoltf ronr N- D- O-& Klpclwcp c~snaeru olaby nr cerocp xanic eg nfb lfau

eg n> ; n0;&; nf &