NCERT पाठ्यपुस्तक पाठ 5 - समांतर श्रेढियाँ, कक्षा 10, गणित Class 10 Notes | EduRev

गणित कक्षा 10

Class 10 : NCERT पाठ्यपुस्तक पाठ 5 - समांतर श्रेढियाँ, कक्षा 10, गणित Class 10 Notes | EduRev

 Page 1


102 xf.kr
5
5.1 Hkwfedk
vkius bl ij vo'; è;ku fn;k gksxk fd izo`Qfr esa] vusd oLrq,¡ ,d fuf'pr izfr:i
(pattern) dk vuqlj.k djrh gaS] tSls fd lwjteq[kh osQ iwQy dh ia[kqfM+;k¡] e/q&dks"k (;k
e/q&NÙks) esa fNnz] ,d HkqV~Vs ij nkus] ,d vuUukl vkSj ,d ikbu dksu (pine cone) ij
l£iy] bR;kfn
vc ge vius nSfud thou esa vkus okys izfr:iksa dh vksj ns[krs gSaA ,sls oqQN
mnkgj.k gSa %
(i) jhuk us ,d in osQ fy, vkosnu fd;k vkSj
mldk p;u gks x;kA mls ;g in ` 8000 osQ
ekfld osru vkSj ` 500 ok£"kd dh osru o`f¼ osQ
lkFk fn;k x;kA mldk osru (` esa) igys o"kZ]
nwljs o"kZ] rhljs o"kZ] bR;kfn osQ fy, Øe'k%
8000, 8500, 9000, . . . gksxkA
(ii) ,d lh<+h osQ MaMksa dh yackb;k¡ uhps ls Åij dh
vksj ,d leku :i ls 2 cm ?kVrh tkrh gSaA
(nsf[k, vko`Qfr 5.1)A lcls uhps okyk MaMk
yackbZ esa 45 cm gSA uhps ls] igys] nwljs]
rhljs] - - - MaMksa dh yackb;k¡ (cm esa) Øe'k%
45, 43, 41, 39, 37, 35, 33 vkSj 31 gSaA
(iii) fdlh cpr ;kstuk esa] dksbZ /ujkf'k izR;sd 3 o"kks± osQ ckn Lo;a dh 
5
4
xquh gks tkrh
lekarj Jsf<+;k¡
vko`Qfr 5.1
2018-19
Page 2


102 xf.kr
5
5.1 Hkwfedk
vkius bl ij vo'; è;ku fn;k gksxk fd izo`Qfr esa] vusd oLrq,¡ ,d fuf'pr izfr:i
(pattern) dk vuqlj.k djrh gaS] tSls fd lwjteq[kh osQ iwQy dh ia[kqfM+;k¡] e/q&dks"k (;k
e/q&NÙks) esa fNnz] ,d HkqV~Vs ij nkus] ,d vuUukl vkSj ,d ikbu dksu (pine cone) ij
l£iy] bR;kfn
vc ge vius nSfud thou esa vkus okys izfr:iksa dh vksj ns[krs gSaA ,sls oqQN
mnkgj.k gSa %
(i) jhuk us ,d in osQ fy, vkosnu fd;k vkSj
mldk p;u gks x;kA mls ;g in ` 8000 osQ
ekfld osru vkSj ` 500 ok£"kd dh osru o`f¼ osQ
lkFk fn;k x;kA mldk osru (` esa) igys o"kZ]
nwljs o"kZ] rhljs o"kZ] bR;kfn osQ fy, Øe'k%
8000, 8500, 9000, . . . gksxkA
(ii) ,d lh<+h osQ MaMksa dh yackb;k¡ uhps ls Åij dh
vksj ,d leku :i ls 2 cm ?kVrh tkrh gSaA
(nsf[k, vko`Qfr 5.1)A lcls uhps okyk MaMk
yackbZ esa 45 cm gSA uhps ls] igys] nwljs]
rhljs] - - - MaMksa dh yackb;k¡ (cm esa) Øe'k%
45, 43, 41, 39, 37, 35, 33 vkSj 31 gSaA
(iii) fdlh cpr ;kstuk esa] dksbZ /ujkf'k izR;sd 3 o"kks± osQ ckn Lo;a dh 
5
4
xquh gks tkrh
lekarj Jsf<+;k¡
vko`Qfr 5.1
2018-19
lekarj Jsf<+;k¡ 103
gSA `8000 osQ fuos'k dh 3, 6, 9 vkSj 12 o"kks± osQ ckn ifjiDork jkf'k;k¡ (#i;ksa esa)
Øe'k%
10000, 12500, 15625, vkSj 19531.25 gSaA
(iv) Hkqtkvksa 1, 2, 3, . . . ek=kdksa (units) okys oxks± esa ek=kd oxks± dh la[;k,¡ (nsf[k,
vko`Qfr 5.2) Øe'k%  1
2
, 2
2
, 3
2
, . . . gSaA
vko`Qfr 5.2
(v) 'kdhyk viuh iq=kh dh xqYyd esa  ` 100 rc Mkyrh gS] tc og ,d o"kZ dh gks tkrh
gS rFkk izR;sd o"kZ blesa ` 50 dh o`f¼ djrh tkrh gSA mlosQ igys] nwljs] rhljs]
pkSFks] --- tUe fnolksa ij mldh xqYyd esa Mkyh xbZ jkf'k;k¡ (#i;ksa esa) Øe'k%
100, 150, 200, 250, . . . gksaxhA
(vi) [kjxks'kksa dk ,d ;qXe vius igys eghus esa iztuu djus osQ ;ksX; ugha gSA nwljs vkSj
izR;sd vkus okys eghus esa os ,d u, ;qXe dk iztuu djrs gSaA izR;sd u;k ;qXe
vius nwljs eghus vkSj izR;sd vkus okys eghus esa ,d u, ;qXe dk iztuu djrk
gS (nsf[k, vko`Qfr 5.3)A ;g ekurs gq, fd fdlh [kjxks'k dh e`R;q ugha gksrh gS]
igys] nwljs] rhljs] - - -] NBs eghus osQ izkjaHk esa [kjxks'kksa osQ ;qXeksa dh la[;k Øe'k%
1, 1, 2, 3, 5 vkSj 8 gksxhA
vko`Qfr 5.3
2018-19
Page 3


102 xf.kr
5
5.1 Hkwfedk
vkius bl ij vo'; è;ku fn;k gksxk fd izo`Qfr esa] vusd oLrq,¡ ,d fuf'pr izfr:i
(pattern) dk vuqlj.k djrh gaS] tSls fd lwjteq[kh osQ iwQy dh ia[kqfM+;k¡] e/q&dks"k (;k
e/q&NÙks) esa fNnz] ,d HkqV~Vs ij nkus] ,d vuUukl vkSj ,d ikbu dksu (pine cone) ij
l£iy] bR;kfn
vc ge vius nSfud thou esa vkus okys izfr:iksa dh vksj ns[krs gSaA ,sls oqQN
mnkgj.k gSa %
(i) jhuk us ,d in osQ fy, vkosnu fd;k vkSj
mldk p;u gks x;kA mls ;g in ` 8000 osQ
ekfld osru vkSj ` 500 ok£"kd dh osru o`f¼ osQ
lkFk fn;k x;kA mldk osru (` esa) igys o"kZ]
nwljs o"kZ] rhljs o"kZ] bR;kfn osQ fy, Øe'k%
8000, 8500, 9000, . . . gksxkA
(ii) ,d lh<+h osQ MaMksa dh yackb;k¡ uhps ls Åij dh
vksj ,d leku :i ls 2 cm ?kVrh tkrh gSaA
(nsf[k, vko`Qfr 5.1)A lcls uhps okyk MaMk
yackbZ esa 45 cm gSA uhps ls] igys] nwljs]
rhljs] - - - MaMksa dh yackb;k¡ (cm esa) Øe'k%
45, 43, 41, 39, 37, 35, 33 vkSj 31 gSaA
(iii) fdlh cpr ;kstuk esa] dksbZ /ujkf'k izR;sd 3 o"kks± osQ ckn Lo;a dh 
5
4
xquh gks tkrh
lekarj Jsf<+;k¡
vko`Qfr 5.1
2018-19
lekarj Jsf<+;k¡ 103
gSA `8000 osQ fuos'k dh 3, 6, 9 vkSj 12 o"kks± osQ ckn ifjiDork jkf'k;k¡ (#i;ksa esa)
Øe'k%
10000, 12500, 15625, vkSj 19531.25 gSaA
(iv) Hkqtkvksa 1, 2, 3, . . . ek=kdksa (units) okys oxks± esa ek=kd oxks± dh la[;k,¡ (nsf[k,
vko`Qfr 5.2) Øe'k%  1
2
, 2
2
, 3
2
, . . . gSaA
vko`Qfr 5.2
(v) 'kdhyk viuh iq=kh dh xqYyd esa  ` 100 rc Mkyrh gS] tc og ,d o"kZ dh gks tkrh
gS rFkk izR;sd o"kZ blesa ` 50 dh o`f¼ djrh tkrh gSA mlosQ igys] nwljs] rhljs]
pkSFks] --- tUe fnolksa ij mldh xqYyd esa Mkyh xbZ jkf'k;k¡ (#i;ksa esa) Øe'k%
100, 150, 200, 250, . . . gksaxhA
(vi) [kjxks'kksa dk ,d ;qXe vius igys eghus esa iztuu djus osQ ;ksX; ugha gSA nwljs vkSj
izR;sd vkus okys eghus esa os ,d u, ;qXe dk iztuu djrs gSaA izR;sd u;k ;qXe
vius nwljs eghus vkSj izR;sd vkus okys eghus esa ,d u, ;qXe dk iztuu djrk
gS (nsf[k, vko`Qfr 5.3)A ;g ekurs gq, fd fdlh [kjxks'k dh e`R;q ugha gksrh gS]
igys] nwljs] rhljs] - - -] NBs eghus osQ izkjaHk esa [kjxks'kksa osQ ;qXeksa dh la[;k Øe'k%
1, 1, 2, 3, 5 vkSj 8 gksxhA
vko`Qfr 5.3
2018-19
104 xf.kr
mijksDr mnkgj.kksa esa] ge oqQN izfr:i ns[krs gSaA oqQN esa] ge ns[krs gSa fd
mÙkjksÙkj in vius ls igys in esa ,d fLFkj la[;k tksM+us ls izkIr gksrs gSa_ oqQN esa
;s in vius ls igys in dks ,d fuf'pr la[;k ls xq.kk djosQ izkIr gksrs gSa rFkk oqQN
vU; esa ge ;g ns[krs gSa fd ;s Øekxr la[;kvksa osQ oxZ gSa] bR;kfnA
bl vè;k; esa] ge buesa ls ,d izfr:i dk vè;;u djsaxs ftlesa mÙkjksÙkj in
vius ls igys inksa esa ,d fuf'pr la[;k tksM+us ij izkIr fd, tkrs gSaA ge ;g Hkh ns[ksaxs
fd buosQ nosa in vkSj n Øekxr inksa osQ ;ksx fdl izdkj Kkr fd, tkrs gSa rFkk bl Kku
dk iz;ksx oqQN nSfud thou dh leL;kvksa dks gy djus esa djsaxsA
5.2 lekarj Jsf<+;k¡
la[;kvksa dh fuEufyf[kr lwfp;ksa (lists) ij fopkj dhft,%
(i) 1, 2, 3, 4, . . .
(ii) 100, 70, 40, 10, . . .
(iii)–3, –2, –1, 0, . . .
(iv) 3, 3, 3, 3, . . .
(v) –1.0, –1.5, –2.0, –2.5, . . .
lwph dh izR;sd la[;k ,d in (term) dgykrk gSA
mijksDr lwfp;ksa esa ls izR;sd lwph esa] ;fn vkidks ,d in fn;k gks] rks D;k vki
mldk vxyk in fy[k ldrs gSa\ ;fn gk¡] rks vki ,slk oSQls djsaxs\ 'kk;n] fdlh
izfr:i ;k fu;e dk vuqlj.k djrs gq,] vki ,slk djsaxsA vkb,] mijksDr lwfp;ksa dks ns[ksa
vkSj buesa lac¼ fu;e dks fy[ksaA
(i) esa izR;sd in vius fiNys in ls 1 vf/d gSA
(ii) esa izR;sd in vius fiNys in ls 30 de gSA
(iii) esa izR;sd in vius fiNys in esa 1 tksM+us ls izkIr gksrk gSA
(iv) esa lHkh in 3 gSa] vFkkZr~ izR;sd in vius fiNys in esa 'kwU; tksM+dj (;k mlesa
ls 'kwU; ?kVk dj izkIr gksrk gSA)
(v) esa izR;sd in vius fiNys in esa – 0.5 tksM+dj (vFkkZr~ mlesa ls 0.5 ?kVkdj)
izkIr gksrk gSA
mijksDr lwfp;ksa esa ls izR;sd esa ge ns[krs gSa fd mÙkjksÙkj inksa dks buls igys inksa
2018-19
Page 4


102 xf.kr
5
5.1 Hkwfedk
vkius bl ij vo'; è;ku fn;k gksxk fd izo`Qfr esa] vusd oLrq,¡ ,d fuf'pr izfr:i
(pattern) dk vuqlj.k djrh gaS] tSls fd lwjteq[kh osQ iwQy dh ia[kqfM+;k¡] e/q&dks"k (;k
e/q&NÙks) esa fNnz] ,d HkqV~Vs ij nkus] ,d vuUukl vkSj ,d ikbu dksu (pine cone) ij
l£iy] bR;kfn
vc ge vius nSfud thou esa vkus okys izfr:iksa dh vksj ns[krs gSaA ,sls oqQN
mnkgj.k gSa %
(i) jhuk us ,d in osQ fy, vkosnu fd;k vkSj
mldk p;u gks x;kA mls ;g in ` 8000 osQ
ekfld osru vkSj ` 500 ok£"kd dh osru o`f¼ osQ
lkFk fn;k x;kA mldk osru (` esa) igys o"kZ]
nwljs o"kZ] rhljs o"kZ] bR;kfn osQ fy, Øe'k%
8000, 8500, 9000, . . . gksxkA
(ii) ,d lh<+h osQ MaMksa dh yackb;k¡ uhps ls Åij dh
vksj ,d leku :i ls 2 cm ?kVrh tkrh gSaA
(nsf[k, vko`Qfr 5.1)A lcls uhps okyk MaMk
yackbZ esa 45 cm gSA uhps ls] igys] nwljs]
rhljs] - - - MaMksa dh yackb;k¡ (cm esa) Øe'k%
45, 43, 41, 39, 37, 35, 33 vkSj 31 gSaA
(iii) fdlh cpr ;kstuk esa] dksbZ /ujkf'k izR;sd 3 o"kks± osQ ckn Lo;a dh 
5
4
xquh gks tkrh
lekarj Jsf<+;k¡
vko`Qfr 5.1
2018-19
lekarj Jsf<+;k¡ 103
gSA `8000 osQ fuos'k dh 3, 6, 9 vkSj 12 o"kks± osQ ckn ifjiDork jkf'k;k¡ (#i;ksa esa)
Øe'k%
10000, 12500, 15625, vkSj 19531.25 gSaA
(iv) Hkqtkvksa 1, 2, 3, . . . ek=kdksa (units) okys oxks± esa ek=kd oxks± dh la[;k,¡ (nsf[k,
vko`Qfr 5.2) Øe'k%  1
2
, 2
2
, 3
2
, . . . gSaA
vko`Qfr 5.2
(v) 'kdhyk viuh iq=kh dh xqYyd esa  ` 100 rc Mkyrh gS] tc og ,d o"kZ dh gks tkrh
gS rFkk izR;sd o"kZ blesa ` 50 dh o`f¼ djrh tkrh gSA mlosQ igys] nwljs] rhljs]
pkSFks] --- tUe fnolksa ij mldh xqYyd esa Mkyh xbZ jkf'k;k¡ (#i;ksa esa) Øe'k%
100, 150, 200, 250, . . . gksaxhA
(vi) [kjxks'kksa dk ,d ;qXe vius igys eghus esa iztuu djus osQ ;ksX; ugha gSA nwljs vkSj
izR;sd vkus okys eghus esa os ,d u, ;qXe dk iztuu djrs gSaA izR;sd u;k ;qXe
vius nwljs eghus vkSj izR;sd vkus okys eghus esa ,d u, ;qXe dk iztuu djrk
gS (nsf[k, vko`Qfr 5.3)A ;g ekurs gq, fd fdlh [kjxks'k dh e`R;q ugha gksrh gS]
igys] nwljs] rhljs] - - -] NBs eghus osQ izkjaHk esa [kjxks'kksa osQ ;qXeksa dh la[;k Øe'k%
1, 1, 2, 3, 5 vkSj 8 gksxhA
vko`Qfr 5.3
2018-19
104 xf.kr
mijksDr mnkgj.kksa esa] ge oqQN izfr:i ns[krs gSaA oqQN esa] ge ns[krs gSa fd
mÙkjksÙkj in vius ls igys in esa ,d fLFkj la[;k tksM+us ls izkIr gksrs gSa_ oqQN esa
;s in vius ls igys in dks ,d fuf'pr la[;k ls xq.kk djosQ izkIr gksrs gSa rFkk oqQN
vU; esa ge ;g ns[krs gSa fd ;s Øekxr la[;kvksa osQ oxZ gSa] bR;kfnA
bl vè;k; esa] ge buesa ls ,d izfr:i dk vè;;u djsaxs ftlesa mÙkjksÙkj in
vius ls igys inksa esa ,d fuf'pr la[;k tksM+us ij izkIr fd, tkrs gSaA ge ;g Hkh ns[ksaxs
fd buosQ nosa in vkSj n Øekxr inksa osQ ;ksx fdl izdkj Kkr fd, tkrs gSa rFkk bl Kku
dk iz;ksx oqQN nSfud thou dh leL;kvksa dks gy djus esa djsaxsA
5.2 lekarj Jsf<+;k¡
la[;kvksa dh fuEufyf[kr lwfp;ksa (lists) ij fopkj dhft,%
(i) 1, 2, 3, 4, . . .
(ii) 100, 70, 40, 10, . . .
(iii)–3, –2, –1, 0, . . .
(iv) 3, 3, 3, 3, . . .
(v) –1.0, –1.5, –2.0, –2.5, . . .
lwph dh izR;sd la[;k ,d in (term) dgykrk gSA
mijksDr lwfp;ksa esa ls izR;sd lwph esa] ;fn vkidks ,d in fn;k gks] rks D;k vki
mldk vxyk in fy[k ldrs gSa\ ;fn gk¡] rks vki ,slk oSQls djsaxs\ 'kk;n] fdlh
izfr:i ;k fu;e dk vuqlj.k djrs gq,] vki ,slk djsaxsA vkb,] mijksDr lwfp;ksa dks ns[ksa
vkSj buesa lac¼ fu;e dks fy[ksaA
(i) esa izR;sd in vius fiNys in ls 1 vf/d gSA
(ii) esa izR;sd in vius fiNys in ls 30 de gSA
(iii) esa izR;sd in vius fiNys in esa 1 tksM+us ls izkIr gksrk gSA
(iv) esa lHkh in 3 gSa] vFkkZr~ izR;sd in vius fiNys in esa 'kwU; tksM+dj (;k mlesa
ls 'kwU; ?kVk dj izkIr gksrk gSA)
(v) esa izR;sd in vius fiNys in esa – 0.5 tksM+dj (vFkkZr~ mlesa ls 0.5 ?kVkdj)
izkIr gksrk gSA
mijksDr lwfp;ksa esa ls izR;sd esa ge ns[krs gSa fd mÙkjksÙkj inksa dks buls igys inksa
2018-19
lekarj Jsf<+;k¡ 105
esa ,d fuf'pr la[;k tksM+dj izkIr fd;k tkrk gSA la[;kvksa dh ,slh lwph dks ;g dgk
tkrk gS fd os ,d lekarj Js<+h (Arithmetic Progression ;k A.P.) cuk jgs gSaA
vr%] ,d lekarj Js<+h la[;kvksa dh ,d ,slh lwph gS ftlesa izR;sd in
(igys in osQ vfrfjDr) vius in esa ,d fuf'pr la[;k tksM+us ij izkIr gksrk
gSA
;g fuf'pr la[;k A.P. dk lkoZ varj (common difference) dgykrh gSA ;kn
jf[k,] ;g lkoZ varj /ukRed] ½.kkRed ;k 'kwU; gks ldrk gSA
vkb, ,d A.P. osQ igys in dks  a
1  
nwljs in dks  a
2
, . . ., nosa in dks  a
n
 rFkk lkoZ
varj dks  d ls O;Dr djsaA rc]  A.P., a
1
, a
2
, a
3
, . . ., a
n  
gks tkrh gSA
vr% a
2
 – a
1
 = a
3
 – a
2
 = . . . = a
n
 – a
n – 1
 = d gSA
A.P. osQ oqQN vU; mnkgj.k fuEufyf[kr gSa %
(a) fdlh LowQy dh izkr%dkyhu lHkk esa ,d iafDr esa [kM+s gq, oqQN fo|k£Fk;ksa dh
Å¡pkb;k¡ ( cm esa ) 147 , 148, 149, . . ., 157  gSaA
(b) fdlh 'kgj esa] tuojh ekl esa fdlh lIrkg esa fy, x, U;wure rkieku (fMxzh
lsfYl;l esa) vkjksgh Øe esa fy[kus ij
– 3.1, – 3.0, – 2.9, – 2.8, – 2.7, – 2.6, – 2.5 gSaA
(c) ` 1000 osQ ,d ½.k esa ls izR;sd ekl 5% ½.k dh jkf'k okfil djus ij 'ks"k
jkf'k;k¡ (` esa) 950] 900] 850] 800] . . ., 50 gSaA
(d) fdlh LowQy }kjk d{kkvksa I ls XII rd osQ lokZf/d vad ikus okys fo|k£Fk;ksa dks
fn, tkus okys udn iqjLdkj (` esa) Øe'k%  200, 250, 300, 350, . . ., 750 gSaA
(e) tc izfr ekl ` 50 dh cpr dh tkrh gS] rks 10 ekl osQ fy,] izR;sd ekl osQ
var esa oqQy cpr dh jkf'k;k¡ (` esa) 50, 100, 150, 200, 250, 300, 350, 400, 450 vkSj
500 gSaA
;g vkiosQ vH;kl osQ fy, NksM+k tk jgk gS fd vki Li"V djsa fd mijksDr esa
izR;sd lwph ,d A.P. D;ksa gSA
vki ;g ns[k ldrs gSa fd
a, a + d, a + 2d, a + 3d, . . .
,d lekarj Js<+h dks fu:fir djrh gS] tgk¡  a igyk in gS vkSj d lkoZ varj gSA bls  A.P.
dk O;kid :i (general form) dgrs gSaA
2018-19
Page 5


102 xf.kr
5
5.1 Hkwfedk
vkius bl ij vo'; è;ku fn;k gksxk fd izo`Qfr esa] vusd oLrq,¡ ,d fuf'pr izfr:i
(pattern) dk vuqlj.k djrh gaS] tSls fd lwjteq[kh osQ iwQy dh ia[kqfM+;k¡] e/q&dks"k (;k
e/q&NÙks) esa fNnz] ,d HkqV~Vs ij nkus] ,d vuUukl vkSj ,d ikbu dksu (pine cone) ij
l£iy] bR;kfn
vc ge vius nSfud thou esa vkus okys izfr:iksa dh vksj ns[krs gSaA ,sls oqQN
mnkgj.k gSa %
(i) jhuk us ,d in osQ fy, vkosnu fd;k vkSj
mldk p;u gks x;kA mls ;g in ` 8000 osQ
ekfld osru vkSj ` 500 ok£"kd dh osru o`f¼ osQ
lkFk fn;k x;kA mldk osru (` esa) igys o"kZ]
nwljs o"kZ] rhljs o"kZ] bR;kfn osQ fy, Øe'k%
8000, 8500, 9000, . . . gksxkA
(ii) ,d lh<+h osQ MaMksa dh yackb;k¡ uhps ls Åij dh
vksj ,d leku :i ls 2 cm ?kVrh tkrh gSaA
(nsf[k, vko`Qfr 5.1)A lcls uhps okyk MaMk
yackbZ esa 45 cm gSA uhps ls] igys] nwljs]
rhljs] - - - MaMksa dh yackb;k¡ (cm esa) Øe'k%
45, 43, 41, 39, 37, 35, 33 vkSj 31 gSaA
(iii) fdlh cpr ;kstuk esa] dksbZ /ujkf'k izR;sd 3 o"kks± osQ ckn Lo;a dh 
5
4
xquh gks tkrh
lekarj Jsf<+;k¡
vko`Qfr 5.1
2018-19
lekarj Jsf<+;k¡ 103
gSA `8000 osQ fuos'k dh 3, 6, 9 vkSj 12 o"kks± osQ ckn ifjiDork jkf'k;k¡ (#i;ksa esa)
Øe'k%
10000, 12500, 15625, vkSj 19531.25 gSaA
(iv) Hkqtkvksa 1, 2, 3, . . . ek=kdksa (units) okys oxks± esa ek=kd oxks± dh la[;k,¡ (nsf[k,
vko`Qfr 5.2) Øe'k%  1
2
, 2
2
, 3
2
, . . . gSaA
vko`Qfr 5.2
(v) 'kdhyk viuh iq=kh dh xqYyd esa  ` 100 rc Mkyrh gS] tc og ,d o"kZ dh gks tkrh
gS rFkk izR;sd o"kZ blesa ` 50 dh o`f¼ djrh tkrh gSA mlosQ igys] nwljs] rhljs]
pkSFks] --- tUe fnolksa ij mldh xqYyd esa Mkyh xbZ jkf'k;k¡ (#i;ksa esa) Øe'k%
100, 150, 200, 250, . . . gksaxhA
(vi) [kjxks'kksa dk ,d ;qXe vius igys eghus esa iztuu djus osQ ;ksX; ugha gSA nwljs vkSj
izR;sd vkus okys eghus esa os ,d u, ;qXe dk iztuu djrs gSaA izR;sd u;k ;qXe
vius nwljs eghus vkSj izR;sd vkus okys eghus esa ,d u, ;qXe dk iztuu djrk
gS (nsf[k, vko`Qfr 5.3)A ;g ekurs gq, fd fdlh [kjxks'k dh e`R;q ugha gksrh gS]
igys] nwljs] rhljs] - - -] NBs eghus osQ izkjaHk esa [kjxks'kksa osQ ;qXeksa dh la[;k Øe'k%
1, 1, 2, 3, 5 vkSj 8 gksxhA
vko`Qfr 5.3
2018-19
104 xf.kr
mijksDr mnkgj.kksa esa] ge oqQN izfr:i ns[krs gSaA oqQN esa] ge ns[krs gSa fd
mÙkjksÙkj in vius ls igys in esa ,d fLFkj la[;k tksM+us ls izkIr gksrs gSa_ oqQN esa
;s in vius ls igys in dks ,d fuf'pr la[;k ls xq.kk djosQ izkIr gksrs gSa rFkk oqQN
vU; esa ge ;g ns[krs gSa fd ;s Øekxr la[;kvksa osQ oxZ gSa] bR;kfnA
bl vè;k; esa] ge buesa ls ,d izfr:i dk vè;;u djsaxs ftlesa mÙkjksÙkj in
vius ls igys inksa esa ,d fuf'pr la[;k tksM+us ij izkIr fd, tkrs gSaA ge ;g Hkh ns[ksaxs
fd buosQ nosa in vkSj n Øekxr inksa osQ ;ksx fdl izdkj Kkr fd, tkrs gSa rFkk bl Kku
dk iz;ksx oqQN nSfud thou dh leL;kvksa dks gy djus esa djsaxsA
5.2 lekarj Jsf<+;k¡
la[;kvksa dh fuEufyf[kr lwfp;ksa (lists) ij fopkj dhft,%
(i) 1, 2, 3, 4, . . .
(ii) 100, 70, 40, 10, . . .
(iii)–3, –2, –1, 0, . . .
(iv) 3, 3, 3, 3, . . .
(v) –1.0, –1.5, –2.0, –2.5, . . .
lwph dh izR;sd la[;k ,d in (term) dgykrk gSA
mijksDr lwfp;ksa esa ls izR;sd lwph esa] ;fn vkidks ,d in fn;k gks] rks D;k vki
mldk vxyk in fy[k ldrs gSa\ ;fn gk¡] rks vki ,slk oSQls djsaxs\ 'kk;n] fdlh
izfr:i ;k fu;e dk vuqlj.k djrs gq,] vki ,slk djsaxsA vkb,] mijksDr lwfp;ksa dks ns[ksa
vkSj buesa lac¼ fu;e dks fy[ksaA
(i) esa izR;sd in vius fiNys in ls 1 vf/d gSA
(ii) esa izR;sd in vius fiNys in ls 30 de gSA
(iii) esa izR;sd in vius fiNys in esa 1 tksM+us ls izkIr gksrk gSA
(iv) esa lHkh in 3 gSa] vFkkZr~ izR;sd in vius fiNys in esa 'kwU; tksM+dj (;k mlesa
ls 'kwU; ?kVk dj izkIr gksrk gSA)
(v) esa izR;sd in vius fiNys in esa – 0.5 tksM+dj (vFkkZr~ mlesa ls 0.5 ?kVkdj)
izkIr gksrk gSA
mijksDr lwfp;ksa esa ls izR;sd esa ge ns[krs gSa fd mÙkjksÙkj inksa dks buls igys inksa
2018-19
lekarj Jsf<+;k¡ 105
esa ,d fuf'pr la[;k tksM+dj izkIr fd;k tkrk gSA la[;kvksa dh ,slh lwph dks ;g dgk
tkrk gS fd os ,d lekarj Js<+h (Arithmetic Progression ;k A.P.) cuk jgs gSaA
vr%] ,d lekarj Js<+h la[;kvksa dh ,d ,slh lwph gS ftlesa izR;sd in
(igys in osQ vfrfjDr) vius in esa ,d fuf'pr la[;k tksM+us ij izkIr gksrk
gSA
;g fuf'pr la[;k A.P. dk lkoZ varj (common difference) dgykrh gSA ;kn
jf[k,] ;g lkoZ varj /ukRed] ½.kkRed ;k 'kwU; gks ldrk gSA
vkb, ,d A.P. osQ igys in dks  a
1  
nwljs in dks  a
2
, . . ., nosa in dks  a
n
 rFkk lkoZ
varj dks  d ls O;Dr djsaA rc]  A.P., a
1
, a
2
, a
3
, . . ., a
n  
gks tkrh gSA
vr% a
2
 – a
1
 = a
3
 – a
2
 = . . . = a
n
 – a
n – 1
 = d gSA
A.P. osQ oqQN vU; mnkgj.k fuEufyf[kr gSa %
(a) fdlh LowQy dh izkr%dkyhu lHkk esa ,d iafDr esa [kM+s gq, oqQN fo|k£Fk;ksa dh
Å¡pkb;k¡ ( cm esa ) 147 , 148, 149, . . ., 157  gSaA
(b) fdlh 'kgj esa] tuojh ekl esa fdlh lIrkg esa fy, x, U;wure rkieku (fMxzh
lsfYl;l esa) vkjksgh Øe esa fy[kus ij
– 3.1, – 3.0, – 2.9, – 2.8, – 2.7, – 2.6, – 2.5 gSaA
(c) ` 1000 osQ ,d ½.k esa ls izR;sd ekl 5% ½.k dh jkf'k okfil djus ij 'ks"k
jkf'k;k¡ (` esa) 950] 900] 850] 800] . . ., 50 gSaA
(d) fdlh LowQy }kjk d{kkvksa I ls XII rd osQ lokZf/d vad ikus okys fo|k£Fk;ksa dks
fn, tkus okys udn iqjLdkj (` esa) Øe'k%  200, 250, 300, 350, . . ., 750 gSaA
(e) tc izfr ekl ` 50 dh cpr dh tkrh gS] rks 10 ekl osQ fy,] izR;sd ekl osQ
var esa oqQy cpr dh jkf'k;k¡ (` esa) 50, 100, 150, 200, 250, 300, 350, 400, 450 vkSj
500 gSaA
;g vkiosQ vH;kl osQ fy, NksM+k tk jgk gS fd vki Li"V djsa fd mijksDr esa
izR;sd lwph ,d A.P. D;ksa gSA
vki ;g ns[k ldrs gSa fd
a, a + d, a + 2d, a + 3d, . . .
,d lekarj Js<+h dks fu:fir djrh gS] tgk¡  a igyk in gS vkSj d lkoZ varj gSA bls  A.P.
dk O;kid :i (general form) dgrs gSaA
2018-19
106 xf.kr
è;ku nhft, fd mijksDr mnkgj.kksa  (a) ls (e) esa] inksa dh la[;k ifjfer (finite) gSA
,slh  A.P.  dks ,d ifjfer A.P. dgrs gSaA vki ;g Hkh ns[k ldrs gSa fd buesa ls izR;sd
A.P. dk ,d vafre in (last term) gSA blh vuqPNsn osQ mnkgj.kksa (i) ls (v) esa nh gqbZ
A.P. ifjfer A.P. ugha gSaA ;s vifjfer A.P.  (Infinite Arithmetic Progressions)
dgykrh gSA ,slh A.P. esa vafre in ugha gksrsA
vc ,d A.P. osQ ckjs esa tkuus osQ fy, vkidks U;wure fdl lwpuk dh vko';drk
gksrh gS\ D;k blosQ izFke in dh tkudkjh i;kZIr gS\ ;k D;k blosQ osQoy lkoZ varj
dh tkudkjh i;kZIr gS\ vki ik,¡xs fd vkidks bu nksuksa vFkkZr~ izFke in a vkSj lkoZ varj
d dh tkudkjh gksuk vko';d gSA
mnkgj.kkFkZ] ;fn izFke in a = 6 gS vkSj lkoZ varj  d = 3 gS rks
 6, 9,12, 15, . . .  A.P.  gSA
rFkk ;fn a = 6 gS vkSj  d = – 3 gS rks
6, 3, 0, –3, . . . A.P. gSA
blh izdkj] tc
a = – 7, d = – 2, rks – 7, – 9, – 11, – 13, . . . A.P. gSA
a =  1.0, d = 0.1, rks 1.0, 1.1, 1.2, 1.3, . . . A.P. gSA
a =  0, d = 1
1
2
, rks 0,  1
1
2
, 3, 4
1
2
, 6, . . . A.P. gSA
a = 2, d = 0, rks  2, 2, 2, 2, . . . A.P. gSA
vr% ;fn vkidks a vkSj d Kkr gksa rks  A.P. fy[k ldrs gSaA bldh foijhr izfØ;k
osQ ckjs esa vki D;k dg ldrs gSa\ vFkkZr~ ;fn vkidks la[;kvksa dh ,d lwph nh gqbZ
gS] rks D;k vki dg ldrs gSa fd ;g ,d  A.P. gS vkSj fiQj blosQ  a  vkSj  d Kkr dj
ldrs gSa\ D;ksafd a izFke in gS] blfy, bls ljyrk ls fy[kk tk ldrk gSA ge tkurs
gSa fd ,d A.P. esa] izR;sd mÙkjksÙkj in vius ls igys in esa  d  tksM+dj izkIr gksrk gSA
vr%] ,d A.P. osQ fy,] mlosQ izR;sd in dks mlls vxys in esa ls ?kVkus ls izkIr d
lHkh inksa osQ fy, ,d gh gksxkA mnkgj.kkFkZ] la[;kvksa dh lwph
6, 9, 12, 15, . . .
osQ fy, gesa izkIr gS% a
2 
– a
1 
= 9 – 6 = 3
a
3
 – a
2
 = 12 – 9 = 3
a
4
 – a
3 
= 15 – 12 = 3
2018-19
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