n!=1 · ...· n n 4!=1 · 2 · 3 · 4 = 24 n! 0! = 1 (n j ) = n! j!(n − j)! 0

n! = 1 · . . . · n
n
4 ! = 1 · 2 · 3 · 4 = 24
n!
0! = 1
¡n¢
j
=
n!
j!(n − j)!
0≤j≤n
¡ 5¢
2
n
¡n¢
n!
n
j
¡n¢
Cnj
j
5!
2!3 !
= 10
=
¡n¢
n!
j
j
n
ak + ak+
1
n
P
+ . . . + an
j=k
j
a 1 + a 2 + . . . + an =
n
P
aj =
j=1
n
P
k=1
ak =
n+
P2
at−2
t=3
(a + b)2 = a2 + 2ab + b2
(a + b)n =
n µ ¶
X
n j n−j
a b
j
j=0
a0 = 1
a n − bn
= (a − b)(an−1 + an−2 b + . . . + abn−2 + bn−1 )
n
X
an−j bj−1
= (a − b)
j=1
aj
j=1
a
1≤j≤n
n
aj = a + (j − 1)d
d
µ
¶
n
X
n−1
aj = n a +
Sn =
·d
2
j=1
a=1
1≤j≤n
n
d=1
n
X
n−1
n(n + 1)
j = n(1 +
· 1) =
=
2
2
j=1
j=1
aj = j
a
µ
¶
n+1
2
1≤j≤n
n
aj = aq j−1
q 6= 1
Sn =
n
X
aq j−1 = a ·
j=1
n+1 ∈ A
1 − qn
1−q
1
n ∈ A
A
A
1≤j≤n
aj = a + (j − 1)d
.
n
X
µ
¶
n−1
aj = n a +
·d
2
j=1
n=1
n+1
n
n
A
n
a
n
P
j=1
n+1
X
j=1
aj
¡
aj = n a +
n−1
2
·d
¢
n
X
¶
µ
n−1
d
aj = (a + nd) + n a +
2
j=1
³
´
n(n − 1)
n
= (n + 1)a + n +
d = (n + 1)a + (n + 1) d
2
2
³
n ´
= (n + 1) a + d
2
=
an+1 +
n
n
2x + 3y + z = 6
x − 2y + z = 0
x+y+z =3
2x + 3y + z = 6
−7y + z = −6
−y + z = 0
y=z=1
y=z
x=1
c
c> 0
c< 0
n
√
n
a>
√
n
b
an > b n
a> b
a+c> b+c
ac > bc
ac < bc
a> b> 0
∗
∗
∗
•
•
−a < x < a
x0 − a < x < x 0 + a
|x| < a
•
|x − x0 | < a
•
a, b
|a + b| ≤ |a| + |b|
b = −y, a = x + y
|x + y| ≥
. a, b
|x| − |y|
•
, |x − 3| + |x + 1| < 9
.
(x − 1)(x − 3)2 (x − 6)3
<0
(x + 4)(x + 5 )2
•
y = ax + b
α
ta n α
a
x
x
(x0 , y0 )
y − y0 = a(x − x0 )
x0 6= x1
(x1 , y1 )
•
a
y = a(x − x0 ) + y0
(x0 , y0 )
y − y0 =
•
y1 − y0
(x − x0 )
x 1 − x0
y1 − y0
x1 − x 0
x
y = a 2 x + b2
1
a1 = −
a2
(x0 , y0 )
(x − x0 )2 + (y − y0 )2 = r2
y = a1 x + b1
a2 6= 0
r
a1 6= 0
•
•
a 6= 1
a>0
x, y
ax+y = ax ay
a−x =
1
ax
(ax )y = axy
lo g a x
a
x>0
lo g a x
x=a
y
lo g a x = y
x, y > 0
lo g a (xy) = lo g a x + lo g a y
µ ¶
1
lo g a
= − lo g a x
x
lo g a xy = y lo g a x
a, b > 0
lo g a x = lo g a b · lo g b x.
α
'
6A $
1
-
_
AB
< AO
&%
O
0,
=α
B
360◦
β
1
α
B
=α
2π
1
β·
π π π π
, , , , π
6 4 3 2
sin(x + 2π) = sin x
c o s(x + 2π) = c o s x
tan(x + π) = tan x
2π
360
a
sin(−x) = − sin x
cos(−x) = cos x
tan(−x) = − tan x
π
2
π
) = cos x
2
1
π
tan(x + ) = −
2
tan x
sin(x +
sin2 x + cos2 x = 1
1
−1
tan2 x =
cos2 x
sin 2x = 2 sin x cos x
cos 2x = cos2 x − sin2 x
x−y
x+y
cos
2
2
x+y
x−y
sin
cos x − cos y = −2 sin
2
2
sin x − sin y = 2 sin
π−t
t + 2π
y = arcsin x
t
sin y = x
cos y = x
tan y = x
√
cos(arcsin x) = 1 − x2
arcsin x
sin t = a
t
0≤y≤π
π
π
− <y<
2
2
−
π
π
≤y≤
2
2
y
y
y = arccos x
y
y = arctan x
1
x
cos(arcsin x)
Z
x
Z 1
Z
Z
cos(arcsin x)
n
p(x) = a0 + a1 x + . . . + an xn =
n
X
aj x j
j=0
an = 1
n
an 6= 0
aj
d e g(p) = n
n
p(x)
p(x) = 0
n
ax2 + bx + c
∆ = b2 − 4ac
√
−b ± ∆
2a
−b
2a
q, r
n = qm + r
∆>0
•
∆=0
•
∆<0
•
0<m<n
•
r<m
0 < m < n
m
g(x)
f (x)
n
r(x)
q(x)
f (x) = q(x)g(x) + r(x)
d e g(r) < m
x3 − x2 + x − 1
x3 + x2
−2x2 + x − 1
−2x2 − 2x
3x − 1
3x + 3
−4
f (x) = x3 − x2 + x − 1
:
x+1
=
r(x) = −4
r(x)
g(x) = x + 1
x2 − 2x + 3
q(x) = x2 − 2x + 3
•
m=kl
•
m>1
k ,l > 1
p(x) = f (x) · g(x)
•
p(x)
deg(g) ≥ 1
deg(f ) ≥ 1
1
2
2
2
x − 1 = (x − 1)(x + 1)
2
x +1
x −1
∗
x +1
∗
2
x + 1 = f (x) · g(x)
deg(f ) ≥ 1
deg(f ) = deg(g) = 1
− ab
f (x)
a 6= 0
∗
2
deg(g) ≥ 1
f (x) = ax + b
2
x2 + 1
x +1
p(x)
p(x) = f1 (x) · · · fk (x)
fj
λ1 · · · λk = 1
n 6= 0
m, n
λj
N = {1, 2, . . .}
N
Z = {0, ±1, ±2, . . .}
m
n
Z
Q
R
A
A
B
A
x
B
A
x
x∈A
x
x∈
/ A
A⊂B
A 6= B
A⊂
6= B
A⊂B
A=B
A⊆B
{x:
x}
Q={x:
A
x
{x∈A:
{x ∈ Z :
−1 ≤ x < 3}
x
{−1, 0, 1, 2}
9
}
x
}