Algebra

1. Identical equation

   $\begin{array}{l}{(a\pm b)}^2=a^2\pm 2ab+b^2\\ {(a\pm b)}^3=a^3\pm 3a^{2}b+3a{b}^2\pm b^3\\ {(a\pm b)}^4=a^4\pm 4a^{3}b+6a^2{b}^2\pm 4a{b}^3+b^4\\ {(a+b+c)}^2=a^2+b^2+c^2+2ab+2ac+2bc\\ {(a+b-c)}^2=a^2+b^2+c^2+2ab-2ac-2bc\\ {(a-b-c)}^2=a^2+b^2+c^2-2ab-2ac+2bc\\ {(a+b+c)}^3=a^3+b^3+c^3+6abc+3(a^{2}b+a{b}^2+b^{2}c+b{c}^2+a{c}^2+c{a}^2)\\ {(a_1+a_2+\mathrm{...}+a_{\mathrm{n}})}^2={a_1}^2+{a_2}^2+...+{a_{\mathrm{n}}}^2+2(a_1{a}_2+a_1{a}_3+...+a_{\mathrm{n-1}}{a}_{\mathrm{n}})\\ a^2-b^2=(a-b)(a+b)\\ a^3+b^3=(a+b)(a^2-ab+b^2)\\ a^3-b^3=(a-b)(a^2+ab+b^2)\\ a^4+b^4={(a^2+b^2)}^2-2a^2{b}^2=(a^2+\sqrt{2}ab+b^2)(a^2-\sqrt{2}ab+b^2)\end{array}$

2. Pow

   $\begin{array}{l}\frac{a^m}{a^n}=a^{m-n}\\ a^m*a^n=a^{m+n}\\ a^{\frac{1}{n}}=\sqrt[n]{a}\\ {(\frac{a}{b})}^m=\frac{a^m}{b^m},(b\ne 0)\\ a^0=1,(a\ne 0)\\ a^{-m}=\frac{1}{a^m},(a\ne 0)\\ a^{\frac{m}{n}}=\sqrt[n]{a^m}\end{array}$

3. Logarithm

   $\begin{array}{l}{\log }_b(N_1{N}_2)={\log }_b|N_1|+{\log }_b|N_2|,(N_1{N}_2>0)\\ {\log }_b\frac{N_1}{N_2}={\log }_b|N_1|-{\log }_b|N_2|,(N_1{N}_2>0)\\ {\log }_b{N}^{\alpha }=\alpha {\log }_{b}N,(N>0)\\ {\log }_b\sqrt[\alpha ]{N}=\frac{1}{\alpha }{\log }_{b}N,(N>0)\\ {\log }_{b}N={\log }_{b}a{\log }_{a}N,(a>0,a\ne 1,N>0)\\ {\log }_{b}a=\frac{1}{{\log }_{a}b},(a>0,a\ne 1)\\ a^{{\log }_{a}b}=b={\log }_a{a}^b\\ {\log }_{\mathrm{10}}N=\lg N(b=10)\end{array}$

4. Sum

   $\begin{array}{l}1+3+5+\mathrm{...}+(2n-3)+(2n-1)=n^2\\ 2+4+6+\mathrm{...}+(2n-2)+2n=n(n+1)\\ 1^2+2^2+3^2+\mathrm{...}+{(n-1)}^2+n^2=\frac{n(n+1)(2n+1)}{6}\\ 1^3+2^3+3^3+\mathrm{...}+{(n-1)}^3+n^3=\frac{n^2{(n+1)}^2}{4}\\ 1^2+3^2+5^2+\mathrm{...}+{(2n-3)}^2+{(2n-1)}^2=\frac{n(4n^2-1)}{3}\\ 1^3+3^3+5^3+\mathrm{...}+{(2n-3)}^3+{(2n-1)}^3=n^2(2n^2-1)\\ 1^4+2^4+3^4+\mathrm{...}+{(n-1)}^4+n^4=\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}\end{array}$

5. Complex number

   $\begin{array}{l}i=\sqrt{-1}\\ i^2=-1\\ i^3=i^2*i=-i\\ i^4=i^3*i=1\\ i^{4n}=1\\ i^{4n+1}=i\\ i^{4n+2}=-1\\ i^{4n+3}=-i\\ (a+bi)+(c+di)=(a+c)+(b+d)i\\ (a+bi)-(c+di)=(a-c)+(b-d)i\\ (a+bi)(a-bi)=a^2+b^2\\ (a+bi)(c+di)=(ac-bd)+(ad+bc)i\\ \frac{a+bi}{c+di}=\frac{a+bi}{c+di}\frac{c-di}{c-di}=\frac{ac+bd}{c^2+d^2}+(\frac{bc-ad}{c^2+d^2})i\end{array}$

6. Arithmetic sequence

   $\begin{array}{l}{a}_2=a_1+d\\ a_3=a_1+2d\\ a_n=a_1+(\mathrm{n}-1)d\\ S_n=a_1+a_2+a_3+\mathrm{...}+a_n=\frac{(a_1+a_n)d}{2}=\frac{[2a_1+(n-1)d]n}{2}\end{array}$

7. Geometric sequence

   $\begin{array}{l}{a}_2=a_{1}q\\ a_3=a_1{q}^2\\ a_n=a_1{q}^{n-1}\\ S_n=a_1+a_2+a_3+\mathrm{...}+a_n=a_1*\frac{q^n-1}{q-1},(q\ne 1)\end{array}$