r`j]w jyv}r

r`j]w jyv}rl[m|b]N]@kt]v]\\l vw\w

n`@g`dkUwr.• (0718005616 @h`\ 0343341702)

mGhr]n\n Exit

r`j]w jyv}r

mGhr]n\n

XY} Er~mk}r~w] p]r]@vn@p`l\vw\w pAsl

@k`l\l[p]t]y @k`LB 03

0718005616 @h`\ 0343341702

Exit

gN]wysm`n\wr @XY\V]10 / 11 @XY\N]

sQh`

r`j]w jyv}r(v]@X\S

gN]w p[h[N[)

mGhr]n\n Exit

(1) hFq]n\v]m. (2) sm`n\wr @XY\V]yk pqyk\ @s””`yn s[wY x`v]wy.

(3) sm`n\wr @XY\V]yk pqvl ekw;v @s””`yn s[wY x`v]wy.

(4) sm`n\wr @XY\V] vl sm`n\wr mE&ny. Exit

sm`n\wr @XY\V] hFq]n\v]m.

(1) 2,4,6,8………………………….. .. .. +2 b#g]n\ ekw;v] a#w.

(2) -25,-23,-21………………………….. .. .. -2 b#g]n\ ekw;v] a#w.

(3) 65,60,55,50………………………….. .. .. -5 b#g]n\ ekw;v] a#w.

ihw a`k`ryt ekm agyk\ ekw;v] n]r~m`Ny

vn @XY\V] sm`n\wr @XY\V] yn[@vn\ hÀ[n\vy].

wvq Pm ekw; vn agy @p`À[an\wry yn[@vn\ hÀ[n\vy].

@p`À[ an\wry d= ps{ pqy - @pr pqy

G#tU

• pYE`n @mn[vt

sm`n\wr @XY\V] h#q]n\v]m g#tU. phw sqhn\ @XY\V] sm`n\wr @XY\V]

@v|q @n`@v|q?(1) 2 5 8 11 ………..

@mh] an\wry sm`n n]s` Py sm`n\wr @XY\V]yk\ @v|.

•pYE`n

@m

n[v

t

an\wry = 5 - 2 =3an\wry = 8 - 5 =3an\wry = 11 - 8=3

• phw e\v` sm`n\wr @XY\V] @v|q @n`@v|q?

(2) 10.25, 10.50, 10.75 ………..

(3) 7½ , -8 , 8 ½, -9 ………..

(4) -3 , 5 , -7 , ………..

(5) 0.2 , 0.5 , 0.8 , ………..

(6) a , a+d ,a+2d ,a+3d….

p]L]w;r

p]L]w;r

p]L]w;r

p]L]w;r

p]L]w;r

(2) 10.25, 10.50, 10.75 sm`n\wr @XY\V]yk\ @v|q @n`@v|q?

• an\wry = 10.50 –-10.25=0.25

• an\wry = 10.75-10.50 =0.25

• emn]s` @my sm`n\wr @XY\V]yk\ @v].

•

• pYE`n @mn[vt

r`j]w jyv}r 0718005616

(3) 7½ , -8 , 8 ½, -9sm`n\wr @XY\V]yk\ @v|q @n`@v|q?

an\wry = -8 - 7 ½, = -15½

an\wry = 8 ½ -(-8) = 8 ½ +8=16½

@my sm`n\wr @XY\V]yk\ @n`@v|.

• pYE`n @mn[vt

(4) -3 , 5 , -7 , sm`n\wr @XY\V]yk\ @v|q @n`@v|q?

an\wry = 5-(-3) = 5+3 =8

an\wry = -7 -5 =-12 @my sm`n\wr @XY\V]yk\ @n`@v].

• pYE`n @mn[vtr`j]w jyv}r 0718005616

(5) 0.2 , 0.5 , 0.8sm`n\wr @XY\V]yk\ @v|q @n`@v|q?an\wry = 0.5 - 0.2 = 0.3 an\wry = 0.8 - 0.5 = 0.3 emn]s` @my sm`n\wr @XY\V]yk\ @v].

• pYE`n @mn[vt

r`j]w jyv}r 0718005616

(6) a , a+d ,a+2d ,a+3d sm`n\wr @XY\V]yk\ @v|q @n`@v|q?

an\wry= a+d - a = d

an\wry=a+2d–(a+d)=a+2d-a-d = d

an\wry=a+3d-(a+2d)=a+3d-a-2d= d

emn]s` @my sm`n\wr @XY\V]yk\ @v|.

• pYE`n @mn[vt

sm`n\wr @XY\V]yk pqyk\ @s`yn s[wYy x`v]wy

@mh] Тn = n vn pqy

n = pq gnn а = m[l\ pqy d =@p`u[ an\wry

Тn = а+(n-1)d

• s[wYy lb` g#n]m g’#tU

• pYE`n @mn[vt

pqyk\ @s`yn s{wYy lb`g#n}m• pLm[ pqy a q n vn pqy T n q

@p`À[ aAwry d q vn sm`n\wr @XY\V]y slkm[.

1 vn pqy T 1 = a2 vn pqy T 2 = a + d3 vn pqy T 3 = a +d +d =a+ 2d4 vn pqy T 4 = a +d +d +d =a+

3d 5 vn pqy T 5 =a +d +d +d +d

=a+4dn vn pqy T n = a + (n-1) d

g’#tU

Tn = a + (n-1) d

pYE`n @mn[vt

g’#tU(1) -2,-4,-6………………………..

sm`n\wr @XY\V]@y\ 10 vn pqy @s`yn\n.

(2) 1/2…,1 1/2, 2 ½1/2, 3 1/2…………………….. sm`n\wr @XY\V]@y\ 7 vn pqy @s`yn\n.

(3) 0.3,0.6,0.9……………………….. sm`n\wr @XY\V]@y\ 6 vn pqy @s`yn\n.

(4) 25,20,15……………………….. sm`n\wr @XY\V]@y\ 9 vn pqy @s`yn\n.

pYE`n @mn[vt

p]L]w;r

p]L]w;r

p]L]w;r

p]L]w;r

Tn = a + (n-1) d a`XY]w g#tU qn\n` pqyk\ @sv}m Tn

pLm[ pqy @sv}m a @p`À[ an\wry @sv}m d pq gNn @sv}m n sAk}r~ g#tU

• pYE`n @mn[vt

(1) -2,-4,-6……………………….. sm`n\wr @XY\V]@y\ 10 vn pqy @s`yn\n.a =-2d = -4-(-2)=-4+2=-2

T10 =?n =10

Tn = a + (n-1) d

T10 = -2 + (10-1) -2

T10 = -2 + 9 X -2

T10 = -2 -18

T10 = -20• pYE`n @mn[vt

(2) 1/2…,1 1/2, 2 ½1/2, 3 1/2…………………….. sm`n\wr @XY\V]@y\7 vn pqy @s`yn\n.a =1/2

d = 1 ½ - ½ =1

T7 =?

n =7

Tn = a + (n-1) d

T7 = ½ + (7-1) 1

T7 = ½ + (6) 1

T7 = ½ + 6

T7 = 6 ½

• pYE`n @mn[vt

(3) 0.3,0.6,0.9……………………….. sm`n\wr @XY\V]@y\ 6 vn pqy @s`yn\n.a =0.3

d = 0.6-0.3=0.3

T6 =?

n =6

• pYE`n @mn[vt

Tn = a + (n-1) d

T6 = 0.3 + (6-1) 0.3

T6 = 0.3 + 5 X 0.3

T6 = 0.3 +1.5

T6 = 1.8

(4) 25,20,15……………………….. sm`n\wr @XY\V]@y\ 9 vn pqy @s`yn\n.a = 25

d = 20-25 = -5

T9 =?

n =9

• pYE`n @mn[vt

Tn = a + (n-1) d

T9 = 25 + (9-1) -5

T9 = 25 + 8 X -5

T9 = 25 - 40

T9 = -15

(2) pLm[ pqy @sv}m(1) 10 vn pqy -20 q

@p`À[ an\wry -2 q v{ @XY\V]@y\ 1 vn pqy @s`yn\n.

(2) 7vn pqy 6 1/2 q @p`À[ an\wry 1 q v{ @XY\V]@y\ 1 vn pqy @s`yn\n.

(3) 6 vn pqy 1.8 q @p`À[ an\wry 0.3 q v{ @XY\V]@y\ 1 vn pqy @s`yn\n.

(4) 9 vn pqy -15 q @p`À[ an\wry -5 q v{ @XY\V]@y\ 1 vn pqy @s`yn\n.

p]L]w;r

p]L]w;r

p]L]w;r

p]L]w;r

•pYE`n

@m

n[v

t

(1) 10 vn pqy -20 q @p`À[ an\wry -2 q v{ @XY\V]@y\ 1 vn pqy @s`yn\n.a =?d = -2T10 =-20n =10

Tn = a + (n-1) dT10 = a + (10-1) -2-20= a + 9 X -2-20= a – 18

-20+18 = a -2=a • pYE`n @mn[vt

(2) 7vn pqy 6 1/2 q @p`À[ an\wry 1 q v{ @XY\V]@y\ 1 vn pqy @s`yn\n.a =?d = 1T7 = 6 ½

n =7

Tn = a + (n-1) d

T7 = a + (7-1) 1

6 ½ = a + 6 X 16 ½ = a + 6

6 ½ - 6 = a ½ =a

• pYE`n @mn[vt

(3) 6 vn pqy 1.8 q @p`À[ an\wry 0.3 q v{ @XY\V]@y\ 1 vn pqy @s`yn\n.a =?

d = 0.3T6 =1.8n =6

Tn = a + (n-1) dT6 = a + (6-1) 0.31.8= a +5 X 0.31.8= a +1.5

1.8 -1.5= a 0.3 = a • pYE`n @mn[vt

(4) 9 vn pqy -15 q @p`À[ an\wry -5 q v{ @XY\V]@y\ 1 vn pqy @s`yn\n.

a =?d = -5T9 = -15

n =9

Tn = a + (n-1) d

T9 = a + (9-1) -5

-15= a +8 X -5-15= a -40

-15+40= a 25 = a

• pYE`n @mn[vt

(3) @p`À[ an\wry @sv}m(1) 1 vn pqy -2 q 10 vn pqy-20

q v{ @XY\V]@y\ @p`À[ an\wry @s`yn\n.

(2) 7 vn pqy 6 1/2 q 1 vn pqy 1/2 q v{ @XY\V]@y\ @p`À[ an\wry @s`yn\n.

(3) 6 vn pqy 1.8 q 1 vn pqy 0.3 q v{ @XY\V]@y\ @p`À[ an\wry @s`yn\n.

(4) 1 vn pqy 25 q 9 vn pqy-15 q v{ @XY\V]@y\ @p`À[ an\wry @s`yn\n.

p]L]w;r

p]L]w;r

p]L]w;r

p]L]w;r

• pYE`n @mn[vt

(1) 1 vn pqy -2 q 10 vn pqy-20 q v{ @XY\V]@y\ @p`À[ an\wry @s`yn\n.

a =-2d = ?T10 = -20

n =10

Tn = a+ (n-1) d

T10 = -2+(10-1) d

-20 = -2+ 9 X d-20 = -2+ 9d-20-2 = 9d-18 = 9d-18 ÷9=d-2=d • pYE`n @mn[vt

(2) 7 vn pqy 6 1/2 q 1 vn pqy 1/2 q v{ @XY\V]@y\ @p`À[ an\wry @s`yn\n.a = ½ d = ?T7 = 6 ½ n =7

Tn = a+ (n-1) dT7 = ½ +(7-1) d6 ½ = ½ + 6 X d6 ½ = ½ + 6d6½-½= 6d 6 = 6d6 ÷6 =d 1 =d• pYE`n @mn[vt

(3) 6 vn pqy 1.8 q 1 vn pqy 0.3 q v{ @XY\V]@y\ @p`À[ an\wry @s`yn\n.a = 0.3 d = ?T6 = 1.8

n =6

Tn = a+ (n-1) dT6 = 0.3 +(6-1)

d1.8 = 0.3 + 5 X

d1.8 = 0.3 + 5d1.8-0.3 = 5d 1.5 = 5d1.5 ÷5 = d 0.3 = d

• pYE`n @mn[vt

(4) 1 vn pqy 25 q 9 vn pqy-15 q v{ @XY\V]@y\ @p`À[ an\wry @s`yn\n.

a = 25 d = ?T9= -15

n =9

Tn = a+ (n-1) dT9 = 25 +(9-1)

d-15 = 25 + 8 X d-15 = 25 + 8d-15-25 = 8d 40 = 8d40 ÷8 = d 5 = d

• pYE`n @mn[vt

(4) pq gNn @sv}m(1) pLm[ pqy -2 q @p`À[ aAwry -

2 q v[ @XY\V]@y\ -20 vn\@n\ k}vn pqyq ?’.

(2) pLm[ pqy 1/2 q @p`À[ aAwry 1 q v[ @XY\V]@y\ 6 1/2 vn\@n\ k}vn pqyq ?’.

(3) pLm[ pqy 0.3 q @p`À[ aAwry 0.3 q v[ @XY\V]@y\ 1.8 vn\@n\ k}vn pqyq ?’.

(4) pLm[ pqy 25 q @p`À[ aAwry -5 q v[ @XY\V]@y\ -15 vn\@n\ k}vn pqyq ?’.

p]L]w;r

p]L]w;r

p]L]w;r

p]L]w;r

• pYE`n @mn[vt

sm`n\wr @XY\V]yk pqvl ekw;v @s`yn s[wYy x`v]wy

sn = n (a+l) 2

sn=n{2a+(n-1)d} 2

sn = pq n vl ekw;v

a = m[l\ pqyl = avsn\ pqyd = @p`À[ aAwry

s[wYy 1 lb`g#n}ms[wYy 2 lb`g#n}m

• pYE`n @mn[vt

g’#tUg’#tU

sn=n (a+l) s[wYy

lb`g#n}m 2plm[ pqy a q pq n vl ekw;v sn q avsn\ pqy l q vn @XY\V]y slkm[.

ev]t @XY\V]y

a,a+d,a+2d ………l-2d,l-d,l @v|.

@mm @XY\V]@y\ ekw;v sn nm|

Sn= a+(a+d)+(a+2d) ……+(l-2d)+(l-d)+l (a+l)

(a+l)(a+l)

• pYE`n @mn[vt

wvq @mh’] pq n a#w’] b#v]n\ a+l pq n/2 k\ a#w.

em n]s` pq n vl ekw;v

sn = n (a+l) 2

g’#tU• pYE`n @mn[vt

sn=n {2a+(n-1)d} s[wYy lb`g#n}m 2

sn=n (a+l) s[wYy slkm[. 2@mm @XY\V]y pq n a#w\nm| avs`n

pqy l vn pqy @v|. enm| l = a + (n -1) d @v|.

ev]t sn=n {a+a+(n-1)d} 2

sn=n {2a+(n-1)d} 2

g’#tU

• pYE`n @mn[vt

s n=n {2a+(n-1)d} s[wYy x`v]wy 2(1) q} a#w] pq gnnk ekw;v @sv}m sn

(2) pLm[ pqy @sv}m a

(3) @p`À[ an\wry @sv}m d

(4) pq gNn @sv}m n

• pYE`n @mn[vt

(1) q} a#w] pq gnnk ekw;v @sv}m sn

(1) 2,4,6,8….. @XY\V]@y\ pq 8 k ekw;v @s`yn\n.

(2) 0.2,0.5,0.8 ….. @XY\V]@y\ pq 7 k ekw;v @s`yn\n.

(3) ½ , ¾ ,1….. @XY\V]@y\ pq 5 k ekw;v @s`yn\n.

p]L]w;r

p]L]w;r

p]L]w;r

• pYE`n @mn[vt

(1) pLm[ pqy -2 q @p`À[ aAwry -2 q v[ @XY\V]@y\ -20 vn\@n\ k}vn pqyq ?’.

a =-2d = -2Tn = -20

n =?

Tn = a+ (n-1) dTn = -2+(n-1) -

2-20 = -2 -2n +

2-20 = -2n-20 ÷-2 =n 10 =n

• pYE`n @mn[vt

(2) pLm[ pqy 1/2 q @p`À[ aAwry 1 q v[ @XY\V]@y\ 6 1/2 vn\@n\ k}vn

pqyq ?’.a = ½ d = 1Tn = 6 ½

n =?

Tn = a+ (n-1) d

Tn = ½ +(n-1) 1

6 ½ = ½ + n - 1

6 ½ +½ = n 6 =n • pYE`n @mn[vt

(3) pLm[ pqy 0.3 q @p`À[ aAwry 0.3 q v[ @XY\V]@y\ 1.8 vn\@n\ k}vn pqyq ?’.

a = 0.3

d = 0.3

Tn = 1.8

n =?

Tn = a+ (n-1) d1.8 = 0.3+(n-1) 0.31.8 = 0.3 +0.3n-0.31.8 = 0.3n1.8 ÷0.3=n 0.6 =n

• pYE`n @mn[vt

(4) pLm[ pqy 25 q @p`À[ aAwry -5 q v[ @XY\V]@y\ -15 vn\@n\ k}vn pqyq ?’.

• a = 25 • d = -5• Tn = -15 • n =?

Tn = a+ (n-1) d-15 = 25+(n-1) (-5)-15 = 25 -5n+5-15 = 30 -5n-15 -30 = -5n-45 ÷-5 = n 9 = n

• pYE`n @mn[vt

(1) 2,4,6,8….. @XY\V]@y\ pq 8 k ekw;v @s`yn\n.

a = 2

d = 4 – 2 = 2

S8 = ?

n =8

Sn = n {2a+(n-1)d} 2

S8 = 8 {2 x2+(8-1)2}

2

S8 = 4 {4+7x2}

S8 = 4 {4+14}

S8 = 4 x 18

S8 = 72• pYE`n @mn[vt

(2) 0.2,0.5,0.8 ….. @XY\V]@y\ pq 7 k ekw;v @s`yn\n.

a = 0.2

d = 0.5-0.2

= 0.3

S7 = ?

n =7

Sn = n {2a+(n-1)d} 2

S7 = 7{2x0.2+(7-1)0.3}

2

S7 = 3.5 {0.4+6x0.3}

S7 = 3.5 {0.4+1.8}

S7 = 3.5 x 2.2

S7 = 7.7

• pYE`n @mn[vt

(3) ½ , ¾ ,1….. @XY\V]@y\ pq 5 k ekw;v @s`yn\n.

a = ½ d = ¾ – ½

= ¼ S5= ?

n =5

Sn = n {2a+(n-1)d} 2S5=5{2x½ +(5-1) ¼}

2

S5 = 2.5 {1+4x ¼ }

S5 = 2.5 {1+1}

S5 = 2.5 x 2

S5 = 5• pYE`n @mn[vt

(2) pLm[ pqy @sv}m a• (1) sm`n\wr @XY\V]yk pq

8 k ekw;v 72 q @p`À[aAwry 2 q vn @XY\V]@y\ pLm[ pqy @s`yn\n.

• (2) sm`n\wr @XY\V]yk pq 7 k ekw;v 7.7 q @p`À[aAwry 0.3 q vn @XY\V]@y\ pLm[ pqy @s`yn\n.

• (3) sm`n\wr @XY\V]yk pq 5 k ekw;v 5 q @p`À[aAwry ¼ q vn @XY\V]@y\ pLm[ pqy @s`yn\n.

p]L]w;r

p]L]w;r

p]L]w;r

• pYE`n @mn[vt

(1) sm`n\wr @XY\V]yk pq 8 k ekw;v 72 q @p`À[ aAwry 2 q vn @XY\V]@y\ pLm[ pqy @s`yn\n.

a = ?d = 2S8 = 72

n =8

Sn = n {2a+(n-1)d} 272 = 8{2a+(8-1)2} 2 72 = 4 {2a+7x2}72 = 4 {2a+14}72 = 8a +5672-56 = 8a16÷8 = a 2 = a

• pYE`n @mn[vt

(2) sm`n\wr @XY\V]yk pq 7 k ekw;v 7.7 q @p`À[aAwry 0.3 q vn @XY\V]@y\ pLm[ pqy @s`yn\n.

a = ?d = 0.3S7 = 7.7

n =7

Sn = n {2a+(n-1)d} 27.7 = 7{2a+(7-1)0.3}

2

7.7 = 3.5 {2a+6x0.3}

7.7 = 3.5 {2a+1.8}

7.7 = 7a +6.3

7.7-6.3 = 7a

1.4÷7 = a

0.2 = a

• pYE`n @mn[vt

(3) sm`n\wr @XY\V]yk pq 5 k ekw;v 5 q @p`À[aAwry ¼ q vn @XY\V]@y\ pLm[ pqy @s`yn\n.

a = ?d = ¼ S5 = 5

n =5

Sn = n {2a+(n-1)d} 25 = 5{2a+(5-1) ¼ } 2 5 = 2.5 {2a+4x ¼ }5 = 2.5 {2a+1}5 = 5a +2.55-2.5 = 5a2.5÷5 = a 0.5 = a

• pYE`n @mn[vt

(3) @p`À[ an\wry @sv}m d►(1) sm`n\wr @XY\V]yk pq 8 k

ekw;v 72 q pLm[ pqy 2 q vn @XY\V]@y\ @p`À[aAwry @s`yn\n.

►(2) sm`n\wr @XY\V]yk pq 7 k ekw;v 7.7 q pLm[ pqy 0.2 q vn @XY\V]@y\ @p`À[aAwry @s`yn\n.

►(3) sm`n\wr @XY\V]yk pq 5 k ekw;v 5 q pLm[ pqy ½ q vn @XY\V]@y\ @p`À[aAwry @s`yn\n.

p]L]w;r

p]L]w;r

p]L]w;r

• pYE`n @mn[vt

(1) sm`n\wr @XY\V]yk pq 8 k ekw;v 72 q pLm[ pqy 2 q vn @XY\V]@y\ @p`À[aAwry @s`yn\n.

a = 2

d = ?

S8 = 8

n =8

Sn = n {2a+(n-1)d} 272 = 8{2x2+(8-1) d}

2

72 = 4 {4+7d }

72 ÷4 = 4+7d

18 = 4 +7d

18-4 = 7d

14÷7 = d

2 = d• pYE`n @mn[vt

(2) sm`n\wr @XY\V]yk pq 7 k ekw;v 7.7 q pLm[ pqy 0.2 q vn @XY\V]@y\ @p`À[aAwry @s`yn\n.

a = 0.2d = ? S7 = 7.7

n =7

Sn = n {2a+(n-1)d} 27.7 = 7{2x0.2+(7-1) d}

2

7.7 = 3.5 {0.4+6d }

7.7 ÷3.5 = 0.4+6d

2.2 = 0.4 +6d

2.2-0.4 = 6d

1.8÷6 = d

0.3 = d• pYE`n @mn[vt

(3) sm`n\wr @XY\V]yk pq 5 k ekw;v 5 q pLm[ pqy ½ q vn @XY\V]@y\ @p`À[aAwry @s`yn\n.

a = ½

d = ?

S5 = 5

n =5

Sn = n {2a+(n-1)d} 2

5 = 5{2x ½ +(5-1) d}

2

5 = 2.5 {1+4d }

5 ÷2.5 = 1+4d

2 = 1 +4d

2-1 = 4d

1÷4 = d

¼ = d

• pYE`n @mn[vt

(4) pq gNn @sv}m n (1) pLm[ pqy 2 q

@p`À[aAwry 2q vn sm`n\wr @XY\V]@y\ ekw;v 72 vn\@n\ k’”}vn pqyq?

(2) pLm[ pqy 0.2 q @p`À[aAwry 0.3 q vn sm`n\wr @XY\V]@y\ ekw;v 7.7 vn\@n\ k’”}vn pqyq?

(3) pLm[ pqy ½ q @p`À[aAwry ¼ q vn sm`n\wr @XY\V]@y\ ekw;v 5 vn\@n\ k’”}vn pqyq?

p]L]w;r

p]L]w;r

p]L]w;r

• pYE`n @mn[vt

(1) pLm[ pqy 2 q @p`À[aAwry 2q vn sm`n\wr @XY\V]@y\ ekw;v 72 vn\@n\ k’”}vn pqyq?

a = 2

d = 2

Sn = 72

n =?

Sn = n {2a+(n-1)d} 272 =n{2x 2 +(n-1) 2}

2

72x2 = n {4+2n-2 }

144 = n{2+2n}

144 = 2n+2n2 ( 2 ÷ n\)n2 +n-72= 0

n2 +9n-8n-72= 0

n(n+9) -8(n+9)=0

(n+9) (n-8)=0

n+9=0 @h`\ n-8=0

n=-9 @h`\ n=8

pq gNnk\ - v]y @n`h#k] b#v]n\

n=8

• pYE`n @mn[vt

(2) pLm[ pqy 0.2 q @p`À[aAwry 0.3 q vn sm`n\wr @XY\V]@y\ ekw;v 7.7 vn\@n\ k’”}vn pqyq?

a = 0.2d = 0.3 Sn = 7.7n =?Sn = n {2a+(n-1)d} 27.7 =n{2x0.2 +(n-1) 0.3} 2 7.7x2 = n{0.4+0.3n-0.3}15.4 = n{0.1+0.3n}

15.4 = 0.1n+0.3n2

0.3n2 +0.1n-15.4= 0(10x n\ )

3n2 +n-154= 0

3n2 +22n-21n-154= 0

n(3n+22) -7(3n+22)=0

(3n+22) (n-7)=0

3n+22=0 @h`\ n-7=0

n=-22 ÷3 @h`\ n=7


n=8

• pYE`n @mn[vt

(3) pLm[ pqy ½ q @p`À[aAwry ¼ q vn sm`n\wr @XY\V]@y\ ekw;v 5 vn\@n\ k’”}vn pqyq?

a = ½ d = ¼ Sn = 5n =?Sn = n {2a+(n-1)d} 25 =n{2x ½ +(n-1) ¼ } 2 5x2 = n{1+ ¼ n- ¼ }10 = n{ ¾ + ¼ n}

10 = ¾ n+ ¼ n2

n2 +3n-40= 0(4x n\ )

n2 +3n-40= 0

n2 +8n-5n-40= 0

n(n+8) -5(n+8)=0

(n+8) (n-5)=0

n+8=0 @h`\ n-5=0

n=-8 @h`\ n=5


n=5

•pYE`n

@m

n[v

t

s n=n (a+l) s[wYy x`v]wy 2


(2) pLm[ pqy @sv}m a (3) avsn\ pqy @sv}m l(4) pq gNn @sv}mn •

pYE`n

@m

n[v

t


(1) pLm[ pqy 2 q 10 vn pqy 20 q vn sm`n\wr @XY\V]@y\ m[l\ pq 10 @y\ ekw;v @s`yn\n.?

(2) pLm[ pqy 0.5 q 7 vn pqy 1.8 q vn sm`n\wr @XY\V]@y\ m[l\ pq 7 @y\ ekw;v @s`yn\n.?

(3) pLm[ pqy ½ q 5 vn pqy 1 ½ q vn sm`n\wr @XY\V]@y\ m[l\ pq 5 @y\ ekw;v @s`yn\n.?

p]L]w;r

p]L]w;r

p]L]w;r

• pYE`n @mn[vt

(1) pLm[ pqy 2 q 10 vn pqy 20 q vn sm`n\wr @XY\V]@y\ m[l\ pq 10 @y\ ekw;v @s`yn\n.?

a = 2l = 20 S10= ?n =10

S n=n (a+l)

2

S 10=10 (2+20)

2

S 10=5 x 22

S 10= 110

• pYE`n @mn[vt

(2) pLm[ pqy 0.5 q 7 vn pqy 1.8 q vn sm`n\wr @XY\V]@y\ m[l\ pq 7 @y\ ekw;v @s`yn\n.?

a = 0.5l = 1.8 S7= ?

n =7

S n=n (a+l)

2

S 7=7 (0.5+1.8)

2

S 7=3.5 x 2.3

S 7= 8.05

• pYE`n @mn[vt

(3) pLm[ pqy ½ q 5 vn pqy 1 ½ q vn sm`n\wr @XY\V]@y\ m[l\ pq 5 @y\ ekw;v @s`yn\n.?

a = ½ l = 1 ½ S5= ?

n =5

S n=n (a+l)

2

S 5=5 ( ½ +1 ½ )

2

S 5=2.5 x 2

S 5= 5

• pYE`n @mn[vt

(2) pLm[ pqy @sv}m a (1) 10 vn pqy 20 q vn sm`n\

wr @XY\V]@y\ m[l\ pq 10 @y\ ekw;v 110 @v|. pLm[ pqy @s`yn\n.?

(2) 7 vn pqy 1.8 q vn sm`n\wr @XY\V]@y\ m[l\ pq 7@w\ ekw;v 8.05 @v|. pLm[ pqy @s`yn\n.?

(3) 5 vn pqy 1 ½ q vn sm`n\wr @XY\V]@y\ m[l\ pq 5 @y\ ekw;v 5 @v|. pLm[ pqy @s`yn\n.?

p]L]w;r

p]L]w;r

p]L]w;r

• pYE`n @mn[vt

(1) 10 vn pqy 20 q vn sm`n\wr @XY\V]@y\ m[l\ pq 10 @y\ ekw;v 110 @v|. pLm[ pqy @s`yn\n.?

a = ? l = 20 S10= 110

n =10

S n = n (a+l)

2

110 =10 ( a+20 )

2

110 =5 x (a+20)

110 ÷ 5= a+20

22 =a+20

22-20 =a

2 =a

• pYE`n @mn[vt

(2) 7 vn pqy 1.8 q vn sm`n\wr @XY\V]@y\ m[l\ pq 7@w\ ekw;v 8.05 @v|. pLm[ pqy @s`yn\n.?

a = ? l = 1.8S7= 8.05

n =7

S n = n (a+l)

2

8.05 =7 ( a+1.8 )

2

8.05 =3.5x (a+1.8)

8.05÷ 3.5= a+1.8

2.3 =a+1.8

2.3-1.8 =a

0.5 =a• pYE`n @mn[vt

(3) 5 vn pqy 1 ½ q vn sm`n\wr @XY\V]@y\ m[l\ pq 5 @y\ ekw;v 5 @v|. pLm[ pqy @s`yn\n.?

a = ?

l = 1 ½

S5= 5

n =5

S n = n (a+l)

2

5 =5 ( a+1 ½ )

2

5 =2.5x (a+1 ½ )

5÷ 2.5= a+1 ½

2 =a+1 ½

2-1 ½ =a

½ =a• pYE`n @mn[vt

(3) avsn\ pqy @sv}m l (1) 1 vn pqy 2 vn sm`n\wr

@XY\V]@y\ m[l\ pq 10 @y\ ekw;v 110 @v|. avsn\ pqy @s`yn\n.?

(2) 1 vn pqy 0.5 vn sm`n\wr @XY\V]@y\ m[l\ pq 7 @w\ ekw;v 8.05 @v|. avsn\ pqy @s`yn\n.?

(3) 1 vn pqy ½ vn sm`n\wr @XY\V]@y\ m[l\ pq 5 @h\ ekw;v 5 @v|. avsn\ pqy @s`yn\n.?

p]L]w;r

p]L]w;r

p]L]w;r

• pYE`n @mn[vt

(1) 1 vn pqy 2 vn sm`n\wr @XY\V]@y\ m[l\ pq 10 @y\ ekw;v 110 @v|. avsn\ pqy @s`yn\n.?

a = 2 l = ?S10= 110

n =10

S n = n (a+l)

2

110 =10( 2+l )

2

110 =5x (2+l )

110÷ 5= 2+l

22 =2+ l

22-2 = l

20 =l• pYE`n @mn[vt

(2) 1 vn pqy 0.5 vn sm`n\wr @XY\V]@y\ m[l\ pq 7@w\ ekw;v 8.05 @v|. avsn\ pqy @s`yn\n.?

a = 0.5 l = ?S7= 8.05

n =7

S n = n (a+l)

2

8.05 =7( 0.5+l )

2

8.05 =3.5x (0.5+l )

8.05÷ 3.5= 0.5+l

2.3 =0.5+ l

2.3-0.5= l

1.8 =l• pYE`n @mn[vt

(3) 1 vn pqy ½ vn sm`n\wr @XY\V]@y\ m[l\ pq 5 @y\ ekw;v 5 @v|. avsn\ pqy @s`yn\n.?

a = ½ l = ?S5= 5

n =5

S n = n (a+l)

2

5 =5 ( ½ +l )

2

5 =2 ½ x( ½ +l )

5÷ 2 ½ = ½ +l

2 = ½ + l

2- ½ = l

1 ½ =l

• pYE`n @mn[vt

(4) pq gNn @sv}mn• (1) 1 vn pqy 2 avsn\ pqy 20

vn sm`n\wr @XY\V]@y\ ekw;v 110 k\ vn pq gNn @s`yn\n.?

• (2) 1 vn pqy 0.5 avsn\ pqy 1.8 vn sm`n\wr @XY\V]@y\ ekw;v 0.85 k\ vn pq gNn @s`yn\n.?

• (3) 1 vn pqy ½ avsn\ pqy 1 ½ vn sm`n\wr @XY\V]@y\ ekw;v 5 k\ vn pq gNn @s`yn\n.?

p]L]w;r

p]L]w;r

p]L]w;r

• pYE`n @mn[vt

(1) 1 vn pqy 2 avsn\ pqy 20 vn sm`n\wr @XY\V]@y\ ekw;v 110 k\ vn pq gNn @s`yn\n.? a = 2

l = 20

Sn = 110

n =?

S n = n (a+l)

2

110 =n( 2+20)

2

110 x2 =nx 22

220÷ 22= n

10= n

• pYE`n @mn[vt

(2) 1 vn pqy 0.5 avsn\ pqy 1.8 vn sm`n\wr @XY\V]@y\ ekw;v 0.85 k\ vn pq gNn @s`yn\n.?

a = 0.5 l = 1.8Sn = 8.05

n =?

S n = n (a+l)

2

8.05 =n( 0.5+1.8 )

2

8.05 x2 =nx 2.3

16.10÷ 2.3= n

7 = n

• pYE`n @mn[vt

(3) 1 vn pqy ½ avsn\ pqy 1 ½ vn sm`n\wr @XY\V]@y\ ekw;v 5 k\ vn pq gNn @s`yn\n.?

a = ½ l = 1 ½ Sn= 5

n =?

S n = n (a+l)

2

5 =n( ½ +1 ½ )

2

5 x2 =nx 2

10÷ 2= n

5 = n

• pYE`n @mn[vt

sm`n\wr mE&ny

• (1) h#Q]n\v}m• (2) sm`n\wr mE&ny ekk\ @sv}m

• (3) sm`n\wr mE&ny k]h]pyk\ @sv}m• pYE`n @mn[vt

(1) h#Q]n\v}m• q} a#w] pq @qkk\ awr sm`n\wr

@XY\V]yk p]htn @ls m#q pq @sv}m sm`n\wr mE&ny yn[@vn\ hÁ[n\vy].

• enm| • (1) 2…..6 awr sm`n\wr mE&ny 4 @v|.• (2) 3….., .., .., 15 awr sm`n\wr

mE&ny w;n • 6,9,12 @v|. • pYE`n @mn[vt

(2) sm`n\wr mE&ny ekk\ @sv}msm`n\wr mE&ny ekk\ @sv}@m|q}

@qps pq @qk ekw; kr @q@kn\ @bqn[ l#@b|.

• (1) -7… h` -12 awr sm`n\wr mE&ny @s`yn\n.

• (2) 0.2 h` 0.6 awr sm`n\wr mE&ny @s`yn\n.

• (3) x… h` 34 awr sm`n\wr mE&ny 13 @v|. x @s`yn\n.

p]L]w;r

p]L]w;r

p]L]w;r

• pYE`n @mn[vt

(1) -7… h` -12 awr sm`n\wr mE&ny @s`yn\n.

sm`n\wr mE&ny = {(-7)+(-12)} ÷2

=-19 ÷2 =-9.5

• pYE`n @mn[vt

(2) 0.2 h` 0.6 awr sm`n\wr mE&ny @s`yn\n.

• sm`n\wr mE&ny = (0.2+0.6) ÷2

=0.8 ÷2 =0.4

• pYE`n @mn[vt

(3) x… h` 34 awr sm`n\wr mE&ny 13 @v|. x @s`yn\n.

• sm`n\wr mE&ny = (x+34)÷2

13 =(x+34) ÷2

13x2 = x+34 26 = x+34 26-34 = x -8 = x

• pYE`n @mn[vt

(3) sm`n\wr mE&ny k]h]pyk\ @sv}m

(1)7… h` -53 awr sm`n\wr mE&ny 3 k\ @s`yn\n.

(2) ½ h` 3 awr sm`n\wr mE&ny 4 k\ @s`yn\n.

• @mh]q} Tn = a+ (n-1) d s[wYy x`v]w` kr @p`À[aAwry @s`y`@gn pq l]yn[ l#@b|.

p]L]w;r

p]L]w;r

• pYE`n @mn[vt

(1) 7… h` -53 awr sm`n\wr mE&ny 3 k\ @s`yn\n.

7, x, y, z, -53

a = 7

d = ?

T5 = 53

n =5

Tn = a+ (n-1) dT5 = 7+(5-1) d-53-7 = 4d-60 = 4d-60 ÷4 =d -15=d

x = 7+(-15) =7-15 = -8

y = -8+(-15) =-8-15 = -23

z = -23+(-15) =-23-15 = -38

7, -8, -23, -38, -53• pYE`n @mn[vt

(2) ½ h` 3 awr sm`n\wr mE&ny 4 k\ @s`yn\n. ½ ,p,q,r,s, 3

a = ½

d = ?

T6 = 3

n =6

Tn = a+ (n-1) dT6 = ½ +(6-1) d3 - ½ = 5d2 ½ = 5d2 ½ ÷5 =d ½ =d

P= ½ + ½ =1

q= 1 + ½ =1 ½ r = 1 ½ + ½ =2s = 2 + ½ = 2 ½ • pYE`n @mn[vt

(1) -2,-4,-6……………………….. sm`n\wr @XY\V]@y\ 10 vn pqy @s`yn\n.a =-2

d = -4-(-2)=-4+2=-2

T10 =?

n =10

Tn = a + (n-1) d

T10 = -2 + (10-1) -2

T10 = -2 + 9 X -2

T10 = -2 -18

T10 = -20• pYE`n @mn[vt

(2) 1/2…,1 1/2, 2 ½1/2, 3 1/2…………………….. sm`n\wr @XY\V]@y\7 vn pqy @s`yn\n.a =1/2

d = 1 ½ - ½ =1

T7 =?

n =7

T7 = ½ + (7-1) 1

T7 = ½ + (6) 1

T7 = ½ + 6

T7 = 6 ½ • pYE`n @mn[vt

(3) 0.3,0.6,0.9……………………….. sm`n\wr @XY\V]@y\ 6 vn pqy @s`yn\n.

• pYE`n @mn[vt

a =0.3d = 0.6-0.3=0.3T6 =?

n =6

Tn = a + (n-1) d

T6 = 0.3 + (6-1) 0.3

T10 = 0.3 + 5 X 0.3

T10 = 0.3 +1.5

T10 = 1.8

(4) 25,20,15……………………….. sm`n\wr @XY\V]@y\ 9 vn pqy @s`yn\n.

• pYE`n @mn[vt

a = 25d = 20-25 = -5T9 =?

n =9

Tn = a + (n-1) d

T9 = 25 + (9-1) -5

T9 = 25 + 8 X -5

T9 = 25 - 40

T9 = -15

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