1.
n
{\displaystyle \scriptscriptstyle {n}}
(
a
+
b
)
n
=
a
n
+
n
a
n
−
1
b
+
n
(
n
−
1
)
2
a
n
−
2
b
2
+
n
(
n
−
1
)
(
n
−
2
)
3
a
n
−
3
b
3
+
⋯
+
n
(
n
−
1
)
(
n
−
2
)
⋯
(
n
−
r
+
2
)
r
−
1
a
n
−
r
+
1
b
r
−
1
+
⋯
.
{\displaystyle {\begin{aligned}\scriptstyle {(a+b)^{n}=a^{n}+na^{n-1}b}&\scriptstyle {+{\frac {n(n-1)}{\begin{array}{|c}\scriptscriptstyle {2}\\\hline \end{array}}}a^{n-2}b^{2}+{\frac {n(n-1)(n-2)}{\begin{array}{|c}\scriptscriptstyle {3}\\\hline \end{array}}}a^{n-3}b^{3}+\cdots }\\&\scriptstyle {+{\frac {n(n-1)(n-2)\cdots (n-r+2)}{\begin{array}{|c}\scriptscriptstyle {r-1}\\\hline \end{array}}}a^{n-r+1}b^{r-1}+\cdots .}\end{aligned}}}
2.
n
!
=
n
=
1
⋅
2
⋅
3
⋅
4
⋯
(
n
−
1
)
n
.
{\displaystyle \scriptstyle {n!={\begin{array}{|c}\scriptscriptstyle {n}\\\hline \end{array}}=1\cdot 2\cdot 3\cdot 4\cdots (n-1)n.}}
3.
a
x
2
+
b
x
+
c
=
0
{\displaystyle \scriptstyle {ax^{2}+bx+c=0}}
4.
x
2
+
p
x
+
q
=
0
{\displaystyle \scriptstyle {x^{2}+px+q=0}}
p
=
{\displaystyle \scriptstyle {p=}}
q
=
{\displaystyle \scriptstyle {q=}}
5.
l
=
a
+
(
n
−
1
)
d
;
s
=
n
2
(
a
+
l
)
=
n
2
[
2
a
+
(
n
−
1
)
d
]
{\displaystyle \scriptstyle {l=a+(n-1)d;\ s={\frac {n}{2}}(a+l)={\frac {n}{2}}[2a+(n-1)d]}}
6.
l
=
a
r
n
−
1
;
s
=
r
l
−
a
r
−
1
=
a
(
r
n
−
1
r
−
1
{\displaystyle \scriptstyle {l=ar^{n-1};\ s={\frac {rl-a}{r-1}}={\frac {a(r^{n}-1}{r-1}}}}
7.
log
a
b
=
log
a
+
log
b
{\displaystyle \scriptstyle {\log ab=\log a+\log b}}
8.
log
a
b
=
log
a
−
log
b
{\displaystyle \scriptstyle {\log {\frac {a}{b}}=\log a-\log b}}
9.
log
a
n
=
n
log
a
{\displaystyle \scriptstyle {\log a^{n}=n\log a}}
10.
log
a
n
=
1
n
log
a
{\displaystyle \scriptstyle {\log {\sqrt[{n}]{a}}={\frac {1}{n}}\log a}}
11.
log
1
=
0
{\displaystyle \scriptstyle {\log 1=0}}
12.
log
a
a
=
1
{\displaystyle \scriptstyle {\log _{a}a=1}}
13.
log
1
a
=
−
log
a
{\displaystyle \scriptstyle {\log {\frac {1}{a}}=-\log a}}
14.
=
2
π
r
{\displaystyle \scriptstyle {=2\pi r}}
[ 1]
15.
=
π
r
2
{\displaystyle \scriptstyle {=\pi r^{2}}}
![{\displaystyle \scriptstyle {l=a+(n-1)d;\ s={\frac {n}{2}}(a+l)={\frac {n}{2}}[2a+(n-1)d]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc74f6e5bd4b6ab21a0e1d45c5765643233c6f3)
![{\displaystyle \scriptstyle {\log {\sqrt[{n}]{a}}={\frac {1}{n}}\log a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b41cbb8a2921bc9f6c82305037b51cb1246e2d)
