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
% & "'
NS! u ! r
→ +∞
3C
6 12 Ar C x/ 0
J
r #
$ I
b
+
+
+
& && &I
+
& −$
&
NS! u ! r
→ +∞
6 12
$ S ∃
@
$
$
N < @ −D = @ =
≤
$+
A$ + C
$
$+
D
→ +∞
$
= $+
@
+, & >6 @ v$
c
5
,- . /
0
3n
21
21
< Un − 3 =
−3 =
=
do n + 7 > 0
n+7
n+7 n+7
N
>6 - r − $ < ⇔
A
λ ,!
&$
&$
v ⇔ B
− Q M0*
+Q
u ! r #I
Sr − I v
−$ =
#
#λ H
−I v ⇔
D
D
$
r
u!
3F V
>6 - r
C r
r E$ k &r E r
+ N +\
< ⇔ $+
A$E C
>
H
$
4J
$ $
> A − $C
D
xF +0
#
$ $
−$
+$
$
+$
+, I
C r
−D =
r −D <
+$ −
$
&
=
6
$
&
+$ −
<
=
>
$
+$ +
$
R &
D
#
R &
− Q E$ [ 0< fA$C
3C
r −D =
&
I
+$
<
?
&
I
+$
&
<
&
A −& C + I
+, R C u$ # u !
→ ∞ A −&C + $ + I +$
&
] 0Jn u ! −
I
→ +∞
N
&
6
&
# u!
→ +∞
+$
&
= u!
→ +∞ I
vε
$ &
−
I I
+
+$
&
−
I
&
B
D
&
#
+$
$
I
+$
=D
$
/ 1y 0
/ F O _
9
< u$ # I # $
I
$
R
b
$
3C L * ; n %/ 1y
u! = u!
= u!
= D N 0Jn u& # D
+, b 1C N
→ +∞ &
→ +∞ &
&
# A$E$C $& E A&E$C && E ' E A E$C
&
&
→ +∞
&
&
< & $ E I & E ' E A E$C
# A$IE&IEIIE'E ICEA$&E&&EIIE'E &C #
& $& + I & & +
u!
R
→ +∞
+, T C
?
+ A + $C &
A + $C
A + $CA& + $C
+
R
T
&
# u!
→+∞
A + $C &
R R
$
$
&
$
= −
+
_ A _ + $CA _ + &C _ _ + $ _ + &
r #
A + $CA& + $C $
#
R
R
→+∞
T
E u!
$
+
&
$
$
−
+&
+$
4 ! r # $)&
3Cr # $ + I + b +
& && &I
N4f f
$
>6 u !
→ +∞ &
−$
$
&
$
−$
&
−$
u! r #D
→ +∞
+
8 A$C ,A&C
$
$
# +&
&
&&
<
&
$
r = $ + I + b + + & −$
&
&& &I & R
& +$
$
$
$
$
& −$
$
−
0Jn
+
+ +
r = +&
&
&
&& &I
&
& +$
& − $ A$C ?
&
−$
−
& −$
= u!
& +$
?
& −$
→ +∞ &
4
$
4 r # $E & $ −
&
= D H- AIC
_
/ 1y
−$
−
& −$
& +$
A&C
AIC
/ O
9
+, Q
Nf r E$ k &r E r $ # λ
4 Ar E$ r C # Ar r $ C E λ
^H #r r $
AH C - ,
Nf 4;
< H E$ # H E λ 0 0< <
N < r # r$ E Ar& r$C E ArI r&C E ' E Ar r $C A$C # r$EH&E'EH
H&EHIE'EH #
NfA$C ,A&C
r
&
9
=
<
− &C ]A − $C
&
#Ar& r$C A $CE A
− I + &C A&C
&
&
4 r # r$EAr& r$CA $C E
r$
+ Ar & − r $ C
&
u!
→ +∞
p ,
ε BD h 9 δ 6
c
[&H& + A
x ≠ xD
r
&
#
λ
−$
&
+
&
&
&
A & − I + &C
− I + & AIC
&
_
/ 1y
/ O
&
9 8 =,! zAxC _ x
xD
J
6 ∀x !, x − x D v δ S z A x C − p C v ε
!"
] t u ! z Ax C = $T
x →&
&
6 zAxC # x E Rx E b
] t u ! z A x C = $D
x →$
I
6 zAxC # x E &
] t u ! z AxC = I
x →$
+, $
6 zAxC # Ix E $D
+, &
+, I
Ix + &
] t u ! z AxC = I
x+Q
x →∞
I
zAxC #
]t
u ! z AxC = +∞
x+&
x → −& +
+, R
6 ,! zAxC #
+, b
6 ,!
ee ?Y1y
$ X
/
F 3
07 0`_ Y0 /
D
D
9 19
u69 $ ?Y1y
5 0d
NS! /
9
0*
, F
,
*
: 35 D
; n
x& − &
$ + &x − $
$C u !
&x
x→D
I Rx − &
bC u !
a&C u !
x →& x − &
x→ & x& − x + & − &
I x +$ − $− x
xI − $
$− x
u!
QC u !
a IC u !
x
$− 6 x
x→D
x →$ x I − x & + x − $
x→
R
RC u !
x →&
x − Ix − &
x& − R
lC u !
x→
? x− 6 x
$− x
R
c [
#& 6^ #I Ji
0` 4/ h _
; Y1y
u69 & . /0
+, $
6
Ji Y1y
F
F 3 07 3
+, &
W
*
; n
P _
' &' 1 2 3
,
; 1JK
, ≠D
!
45
$ + x _ 0< fx→D
4 →$
$+ x −$
−$
H- u !
=u!
# u!
x
x→D
→$
→$ A − $CA
−$
ui
TC
$+ x −$
=
x
x→D
u!
^ #
60
eAxC # $xE &x&E E x
$
≠D
−$
−$
+
−&
+
+ + $C
#
K
$ + e Ax C − $
x
$ + eAx C − $ eAxC
=> N < u# u !
eAxC
x
x→D
NS!
u# u !
x→D
9
$ + eAx C − $
u ! A $ + &x +
eAxC
x→D
x→D
^ # eAxC _
0<
N
x −$ C #
+
# u!
<
$ + & x − $ + & x + $ + & x I $ + Ix − $ + & x I $ + Ix + $ + & x I $ + Ix R $ + R x − $
$ + & x I $ + Ix R $ + R x − $
H6
& I R
E E #I
& I R
x
&
S →$ ?
4
$ + x& − x =
$
$
= u!
&
x→D x
→$
$+ x + x $ S
−$
A − $C A$ + + & +
#u!
x
x→D
+ & −$ C
→$
$ + x& + x − $
< u# & u !
# & Au ; n C
x
x→D
+, R NS!
=> ^ x #
−
2
x →1 1 − x 1 − 3 x
O 0Jn
9 02 6 35 $)&
T
+, b NS!
x +1 − 2
Lim
9
+, T NS!
=> N/
3
Lim
9
3 x −1
x →1
2 x +1 − 3 8 − x
Lim
x
x →0
9
, &
9 NY
#2
(
)
x +1 −1 + 2 − 3 8 − x
2 x +1 − 3 8 − x
#' #$I)$&
x
x →0
Lim
1
23 0 4
+, Q NS! /
% lim
x →0
sin x
=1
x →0 x
lim
9
2 sin x − sin 2 x
a 3% lim
x →0
tgx − sin x
1 − cos x
%u# lim
3
3
x → 0 2 tg 2 x
x
x
=> %NY # & xA$ 6 xC / ,! & *
Y1y _ @ W 4;
3%NY# ? xA$ 6 xC)6 x / ,! & *
Y1y _ @ W 4;
2
a
sin
%N
1 − cos x
x → +∞
2x − 1 − 4x 2 − 4x − 3
_
*
; n
x+ x+ x − x #
# ' # $)&
<
lim
x → +∞
x+ x
x+ x+ x +
1+
# lim
x x → +∞
1+
x
x
x+ x
+1
x
=,!
5
lim f ( x ) = f ( x 0 )
zAxC ; y 9 xD∈>z ⇔
x→x 0
=,!
zAxC ; y 4/ 9 xD∈>z ⇔
=,!
zAxC ; y
6 7 8&))
x → x 0−
W 9 xD∈>z ⇔
! 9 ) % %:
z AxC =
6 ,!
lim f ( x ) = f ( x 0 )
x&
lim f ( x ) = f ( x 0 )
x → x 0+
8;
$
x
x≠D
x=D
D
.F O ; y 8 ,!
9 xD# D
=> ?Y1y
; _s x&
$)x
&x
z AxC =
6 ,!
?
x=D
.F O ; y 8 ,!
9 xD# &
=> .F
9 4/% WZ 9 xD#&' ,!
zAxC x/ 0
J
02 6 _
x →$
x
= !
x →$ x − $
x →$
!
* .F O
x −$
z AxC =
]
H
6
x
&
W
A − xC
!
# −
x −$
x →$
; y 8 ,!
; y 9 xD#&
x
z AxC = x − $
−
] t ,! 12 6 ; y 4; 6, 4y
=> ] ,! 02 6 ; y 9 x#$
40
! =
; y 9 xD
x≠D
x
&
6 ,!
4 ,!
x ≠$
x =$
!
| A$ − xC{
#
A$ − xC
4; t
x >$
x ≤$
" 35 0h
9 ,! 02 6 1J 19
x −$
z AxC =
x
&
$− x
6
x >$
−$ ≤ x ≤ $
x < −$
&
$
$
$
+
z AxC =
6 ,!
!
x/ 0
I
&
−I
J
+I −x
−x
→$
[ W6 / O ; y 8 ,!
(C t}4, ,! 02 6 _
,!
" 35 0h
; y 9 x#$
$
$
!
!
! " #$ %
!
45
JK 4S xI IxE$# D .F zA &C%zA $C
zA &C%zA $C
, Y1y O ; y 8 ,! 02 6 4; t
+, &
6 %3% V
!
45
JK 4S
& '(
G!
* 3G
<
G!
)
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