U n +1 = U n + d

!" #$%& '
U + U n +1
()U n = U 1 + (n − 1)d
U n = n −1
2
(U 1 + U n )n [2U 1 + (n − 1)d ]n
S n = U 1 + U 2 + ..... + U n =
=
2
2
&
*
* U n +1 = U n q
*
2
n −1
U n = U n −1U n +1
()U n = U 1 q

(

)

U1 q n − 1
()S n = U 1 + U 2 + .... + U n =
q −1
+ +, +, $ ./ 0 ! 12
3

+, - &

6 7

!

45
9 7

=> ?

8

9 0:

45

; 8 !

U 1 = 2;U 8 = 37

12

<19

S n = 5n 2 + 6n

S n ,12

@ / U n 8 12


A B$C

U n = S n − S n −1 # $D E$ .F G U n − U n −1 = (10n + 1) − [10(n − 1) + 1] = 10 #1
H- 12
02 6 ,12
1#$D
+, - I
63
1JK < 7 35 &$ , 96 ; !
L 3 $M
9
;! $ ,6
9
+, $ NO 7 8 $&
9 0: ; 8 !
<
9 0: ; , I P O 8
9
I , Q 35 &R
−1
S?$& # TU
? H 1#
2
5
H 1#
S?$& # $&U
2
+, & NS! /
9 8 !

3 45 7 8 R 0: ; 35 RD P 7 8 R
V 35 $DR N7 8
W/
9 35 &$T
=> X W Y12
WZ S! , ( an ) [ 0< \6 W
a1 + a2 + a3 + a4 = 40
an + an −1 + an − 2 + an −3 = 104

]^ _ /

a1 + an = a2 + an −1 = a3 + an − 2 = a4 + an −3 >6 0< 4 ( a1 + an ) = 144

[ 0< S n =

a1 + an
n = 216
2

H- ÷ ( an ) < ( a1 ) = 7 ,

n = 12 >6 d =2

4a1 + 6d = 40
4a1 + 38d = 104

⇔

a1 = 7
d =2


a1 + an = 36

%


/ 4 8 α 0`1 + sin α ;sin 2 α ;1 + sin 3α - ,
−1
? 6 α #D asin α =
2
+, R
!
45 12
<
9 7 @ / an = 2 n − 5 +, I NS! /

? 1# an − an −1 = 2 , s20 = 320
+, b +
1JK ,R
;
8 !
$UIDb NS! /
0<
X" R
WZ S! , a; a + 2; a + 4; a + 6 N \6
a ( a + 2 )( a + 4 )( a + 6 ) = 19305

(a

2


<

,

NS! ?&D

1# & , O

8

!

45

+ 6a )( a 2 + 6a 2 + 8 ) = 19305

9 0:

1
1
1
1
2
1 1
1
+
+ ... +
+
=

+ + ... +
u1un u2un −1
un −1u2 unu1 u1 + un u1 u2
un

1
1
1
+
+ ... +
u1.u2 u2 .u3
un −1un
u −u
d
d
d
u −u u −u
0< 1?#
+
+ ... +
# 2 1 + 3 2 + ... + n n −1
u1.u2 u2 .u3
un −1un u1.u2
u2 .u3
un −1un

=>$ X" ? #
_

1 1 1 1

1
1 1 1 u −u
− + − + ... +
− = − = n 1
u1 u2 u2 u3
un −1 un u1 un
u1un
1
1
1
n −1
H+
+ ... +
=
u1.u2 u2 .u3
un −1un u1un
=

dS =

( n − 1) d
u1un

S=

n −1
u1un

=> & S =


u − u2
u − un −1
u2 − u1
+ 3
+ ... + n
>6 u2 − u1 = u3 − u2 = ... = un − un −1 = d
u2 − u1
u3 − u2
un − un −1

?

d ( n − 1)
un − u1
un − u1
n −1
=
=
=
d
u1 + un
d u1 + un
d u1 + un

4 ?#

(

35


W

0^ u = a 2 + 6a
[ 0< u = 135; u = −134
• H a 2 + 6a = 135 a = 9 ∨ a = −15
• H a 2 + 6a = −134
G!
1
1
+, T N46 !
< an =
, am =
( m ≠ n ) =2 S! 7 8 !
m
n
8 12
1
a +a
mn + 1
=d
Smn = 1 mn mn =
=> N < a1 =
mn
2
2
+, 6
u1 ; u2 ;...un 46 0< ui > 0∀i = 1, n ,
1
1
1

1
n −1
+
+ ... +
=
$
!
45
u1.u2 u2 .u3
un −1un u1un
1
1
1
n −1
&
!
45
+
+ .... +
=
u1 + u2
u2 + u3
un −1 + un
u1 + un
I

c

)


(

)

;


=> I

d

:
!
JK 0JK
1
1
1
1
1 1
1
A(C
+ ... +
+
)=2
+ + ... +
( u1 + un ) ( +
u1un u2un −1
un −1u2 un u1
u1 u2
un


1
1
1
1
+
+ ... +
+
) HS 46 !"
<
u1un u2un −1
un −1u2 un u1
u +u u +u
u +u u +u
u1 + un = u2 + un −1 = .... = un −1 + u2 = un + u1 >6 0< ?# ( 1 n + 2 n −1 + ... + n −1 2 + n 1 )
u1un
u2un −1
un −1u2
un u1

X" S = ( u1 + un ) (

⇔S=

1 1 1
1
1
1 1 1
1 1
1

+ + +
+ .... +
+ + + #2
+ + ... +
u1 un u2 un −1
un −1 u2 un u1
u1 u2
un

+, -

!

45

46

0c

< S3 n = 3 ( S 2 n − S n )

!"

N - - He# 3 ( S 2 n − S n ) = 3

H- A(C

( 2u + ( 2n − 1) d ) 2n − ( 2u + ( n − 1) d ) n
1


1

2

2

#

2u1 + ( 3n − 1) d 3n
2

= S3 n

+, 6
62!2 O
Sm = Sn
m≠n
!
45
Sm+n = 0
=> Nf W
2u1 + ( m − 1) d m
2u1 + ( n − 1) d n
S m = sn
=
2u1m + ( m 2 − m ) d = 2u1n + ( n 2 − n ) d A(C
2
2
( m − n ) 2u1 + ( m + n − 1) d = 0 >6 m ≠ n ; fA(C 4 2u1 + ( m + n − 1) d = 0 A((C
]^ _ /


Sm+n =

+, - N7
*
=> Nf

2u1 + ( m + n − 1) d

(m + n)

2

8

9 0:
O

7

; 8 12

, S n = 3n − 1

g {an } ,

JK

9 0:


[ 0< u1 = a1 + 2d ; u2 = a1 + 12d ; u3 = a1 + 14d H
N \6

W

u1 + u2 + u3 = 124

] ^ _ / N \6 O

8

!

an = Sn − Sn −1 = ( 3n − 1) − ( 3n −1 − 1)

8 12

an
3.3n −1
=
#I
,
an −1 3.3n − 2
+, - S! I
< 7 35 $&R ,I
9
I%
$I%$b 8 !
=> X W Y u1 = a3 ; u2 = a13 ; u3 = a15


.F

H- S m + n = 0

;

* <
8 !

1,

45 12

,12

an = 3n − 3n −1 = 2.3n −1
3 @#I
* ,0h

i ,/

8

3a1 + 28d = 124 (1)

*

( a1 + 12d )

2


= ( a1 + 2d )( a1 + 14d ) ⇔ 2a1 + 29d = 0 ( 2 )

NfA$C ,A&C
4 a1 = 116; d = −8 H- u1 = 100; u2 = 20; u3 = 4
+, 63
- ,
* L
c 4f JK
j!
=2 S! /
0<
=> X" /
WZ S! , a; aq; aq 2 ; aq 3
[ 0< \6 W
&a@ &a @& k Qa @I k &Q - ,
2 ( aq − 1) = ( a − 2 ) + ( aq 2 − 7 )

6 &%$aQa&Qa

>6 0<

- 0J


2 ( aq − 7 ) = ( aq − 1) + ( aq − 27 ) ?
2

3


H- 3
WZ S! ,
+, - N7
/
9 8 !
9 8 <35 RRl =2 S!

a ( q − 1) = 7
2

4

aq ( q − 1) = 14
2

* V

⇔

a=7
q=2

9 35 bT < 7

3S

JK

8 /


,
2

a
a
2
2
a1 + a1q + ...a1q + ... = 1 = 56 , a1 + a1 q 2 + ... + a1q 2 n + ... = 1 2 = 448
=> N \6 W
1− q
1− q
a1 = 56(1 − q )
3
a1 = 8 (1 + q ) >6 0< 8 (1 + q ) = 56 (1 − q ) q =
, $#$R
[ 0< <
2
2
a1 = 448 (1 − q )
4
n

+, +
< 7 35 Tb - , !
3 $U0K M
3 0Jn !
3 NS! 3
- ,
* 3 45 7
c 35 ITR

?
b a$baRb
+ $laT a& 6^ &aTa$l
+, -

6

*

u1 ; u2 ;....

u1 ( q n − 1)

N - - HN #

!

* m L 3 !
NS! I 0<
8 c 35 &T P 7

45

0K
/ 3S

M

,
JK


8

Sn
S − Sn
= 2n
S 2 n − S n S3 n − S 2 n

qn −1
1
qn −1
q −1
#
=
A(C
=
n
n
2
n
u1 ( q 2 n − 1) u1 ( q n − 1) q − 1 − q + 1 q n ( q n − 1) q
−
q −1
q −1

u1 ( q 2 n − 1) − u1 ( q n − 1)

q 2n − q n
q 2n − q n
1

q −1
#
=
= n A((C
3n
2n
3n
2n
n
2n
n
u1 ( q − 1) u1 ( q − 1) q − q
q (q − q ) q
−
q −1
q −1
H- fA(C ,A((C
4
e ]
A
B
C
+, !
45
tg ; tg ; tg N \6
o- ,
S 6 pa 6 + a 6
2
2
2

q
\6
o- ,
A+C
B
sin
sin
A
C
B
2
2 ⇔ cos 2 B = 2sin B cos A cos C
=> N \6 W
tg + tg = 2tg ⇔
=2
A
C
C
2
2
2
2
2
2
2
cos cos
cos
2
2
2

⇔ 1 + cos B = 1 − cos B + cos A + cos C ⇔ 2 cos B = cos A + cos C
A
B
C
+, !
45
cotg ; cotg ; cotg N \6
o- ,
S3 9
a
2
2
2
3a q
\6
o- ,

He #


=> N \6

W

A+C
B
A+C
sin
cos
2sin

A
C
B
2
2 =
2
cot g + cot g = 2 cot g ⇔
=2
A
C
B
A+C
2
2
2
sin sin
sin
cos
2
2
2
2
A+C
A+C
A+C
A−C
A+C
⇔ sin
cos
= sin

cos
− cos
2
2
2
2
2
1
1
⇔ sin ( A + C ) = ( sin A + sin C ) ⇔ sin B = ( sin A + sin C ) ⇔ a + c = 2b ⇔ ÷a; b; c
2
2
A
B
C
+, !
45
cotg ; cotg ; cotg N \6
o- ,
2
2
2
a 2 ; b2 ; c2 q
\6
o- ,
=> N \6 W
sin ( A + C ) 2 cos B
=
cot gA + cot gC = 2 cot gB ⇔
⇔ 2sin 2 B 2sin 2 B = 2sin sin C cos B

sin A.sin B
sin B
⇔ b 2 = 2ac cos B = a 2 + c 2 − b 2 ⇔ 2b 2 = a 2 + c 2 ⇔ ÷a 2 ; b 2 ; c 2

S3 9



−

( )& * r
/ F O
/ O
8

'

C ]t

∀
+
9 e F
9%
;

12

<

9


7

% 4f% F
_s

*%

@ /

I
S u ! r #I
+Q
→ +∞

r #
3C

Ar C

6 12

r −D #

=>

&

r#


&

&
&

+& +b

+& +b
&
&
v

&

C

! "# $

6 12

Ar C

r#

&

#

&


v

+&

&

"

D

&

#

u! r #$
→ +∞

→ +∞

C= D a

3C ] t u !
→∞

& ?
I

+$

&


+ $C

!" #$ %&% ' (' )
* NS! /

A−&C + I

$
$
+
+
→ +∞ $ & & I

+

C u!

Cr#$

,
C

=D

9

C u!
→ ∞ A −&C +$ + I +$
+ NS! /

9
&
+&
+I
C u!
→ +∞ & & + −

=> C
3C L *

+$

u! @ #D

]t

+$ −

B

]t

+& +R

6 @ !, @ v$
C ]t u ! A
→∞

]t u ! r # D
→ +∞


W Y ,!w
; n

6

J

3C u ! A
→∞

a

3C u !
→ +∞

&

+b −

&

+ +$ −

$
& $& + I & & + + A + $C &
a1C u !
R
+ $)
→ +∞


&

$
>6 0< u !r # $
+$

6 12 Ar C x/ 0

(

a

!

r #

$
#$
+$
$
$
+
+
$&I &IR

+

$
A + $CA + &C


% & "'