常见初等代数公式

文章目录

常见初等代数公式

1 一元二次方程 a x 2 + b x + c ( a ≠ 0 ) ax^2+bx+c(a\not=0) ax2+bx+c(a​=0)

根的判别式 Δ = b 2 − 4 a c \Delta=b^2-4ac Δ=b2−4ac

  • 当 Δ > 0 \Delta\gt 0 Δ>0时,方程有两个相异的实根
  • 当 Δ = 0 \Delta=0 Δ=0时,方程有两个相等的实根
  • 当 Δ < 0 \Delta\lt 0 Δ<0时,方程有共轭复根

求根公式为: x 1 , 2 = − b ± b 2 − 4 a c 2 a x_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a} x1,2​=2a−b±b2−4ac ​​。

2 指数的运算性质

a m ⋅ a n = a m + n a^m\cdot a^n=a^{m+n} am⋅an=am+n

a m a n = a m − n \frac{a^m}{a_n}=a^{m-n} an​am​=am−n

( a m ) n = a m ⋅ n (a^m)^n=a^{m\cdot n} (am)n=am⋅n

( a ⋅ b ) m = a m ⋅ b m (a\cdot b)^m=a^m\cdot b^m (a⋅b)m=am⋅bm

( a b ) m = a m b m (\frac{a}{b})^m=\frac{a^m}{b^m} (ba​)m=bmam​

3 对数的运算性质

若 a y = x a^y=x ay=x,则 y = log ⁡ a x y=\log_a x y=loga​x

log ⁡ a a = 1 , log ⁡ a 1 = 0 , ln ⁡ e = 1 , ln ⁡ 1 = 0 \log_aa=1,\log_a1=0,\ln e=1,\ln1=0 loga​a=1,loga​1=0,lne=1,ln1=0

log ⁡ a ( x ⋅ y ) = log ⁡ a x + log ⁡ a y \log_a(x\cdot y)=\log_ax+\log_ay loga​(x⋅y)=loga​x+loga​y

log ⁡ a x y = log ⁡ a x − log ⁡ a y \log_a\frac{x}{y}=\log_ax-\log_ay loga​yx​=loga​x−loga​y

log ⁡ a x b = b ⋅ log ⁡ a x \log_ax^b=b\cdot\log_ax loga​xb=b⋅loga​x

a log ⁡ a x = x , e ln ⁡ x = x a^{\log_ax}=x,e^{\ln x}=x aloga​x=x,elnx=x

log ⁡ a b = log ⁡ c b log ⁡ c a , log ⁡ a b = 1 log ⁡ b a \log_a b=\frac{\log_c b}{\log_c a},\log_ab=\frac{1}{\log_ba} loga​b=logc​alogc​b​,loga​b=logb​a1​

4 排列组合公式

n ! = n ( n − 1 ) ( n − 2 ) ⋯ 2 ⋅ 1 , 0 ! = 1 n!=n(n-1)(n-2)\cdots 2\cdot 1,0!=1 n!=n(n−1)(n−2)⋯2⋅1,0!=1

排列数: P n m = n ( n − 1 ) ( n − 2 ) ⋯ ( n − m + 1 ) , P n 0 = 1 , P n n = n ! P_n^m=n(n-1)(n-2)\cdots(n-m+1),P^0_n=1,P^n_n=n! Pnm​=n(n−1)(n−2)⋯(n−m+1),Pn0​=1,Pnn​=n!

组合数: C n m = n ( n − 1 ) ( n − 2 ) ⋯ ( n − m + 1 ) m ! = n m ! ( n − m ) ! , , C n 0 = 1 , C n n = 1 C_n^m=\frac{n(n-1)(n-2)\cdots(n-m+1)}{m!} =\frac{n}{m!(n-m)!},,C^0_n=1,C^n_n=1 Cnm​=m!n(n−1)(n−2)⋯(n−m+1)​=m!(n−m)!n​,,Cn0​=1,Cnn​=1

5 常用二项展开及分解公式

( a + b ) 2 = a 2 + 2 a b + b 2 (a+b)^2= a^2+2ab+b^2 (a+b)2=a2+2ab+b2

( a − b ) 2 = a 2 − 2 a b + b 2 (a-b)^2= a^2-2ab+b^2 (a−b)2=a2−2ab+b2

( a + b ) 3 = a 3 + 3 a 2 + 3 a 2 b + 3 a b 2 + b 3 (a+b)^3=a^3+3a^2+3a^2b+3ab^2+b^3 (a+b)3=a3+3a2+3a2b+3ab2+b3

( a − b ) 3 = a 3 − 3 a 2 + 3 a 2 b + 3 a b 2 − b 3 (a-b)^3=a^3-3a^2+3a^2b+3ab^2-b^3 (a−b)3=a3−3a2+3a2b+3ab2−b3

a 2 − b 2 = ( a + b ) ( a − b ) a^2-b^2=(a+b)(a-b) a2−b2=(a+b)(a−b)

a 3 − b 3 = ( a − b ) ( a 2 + a b + b 2 ) a^3-b^3=(a-b)(a^2+ab+b^2) a3−b3=(a−b)(a2+ab+b2)

a 3 + b 3 = ( a + b ) ( a 2 − a b + b 2 ) a^3+b^3=(a+b)(a^2-ab+b^2) a3+b3=(a+b)(a2−ab+b2)

a n − b n = ( a − b ) ( a n − 1 + a n − 2 b + a n − 3 b 2 + ⋯ + b n − 1 ) a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b^2+\cdots+b^{n-1}) an−bn=(a−b)(an−1+an−2b+an−3b2+⋯+bn−1)

( a + b ) n = C n 0 a n + C n 1 a n − 1 b + C n 2 a n − 2 b 2 + ⋯ + C n k a n − k b k + ⋯ C n n b n (a+b)^n=C_n^0a^n+C^1_na^{n-1}b+C_n^2a^{n-2}b^2+\cdots +C_n^ka^{n-k}b^k+\cdots C_n^nb^n (a+b)n=Cn0​an+Cn1​an−1b+Cn2​an−2b2+⋯+Cnk​an−kbk+⋯Cnn​bn

6 常用不等式及其运算性质

如果 a > b a\gt b a>b,则有:

a ± c > b ± c a \pm c>b \pm c a±c>b±c

a c > b c ( c > 0 ) , a c < b c ( c < 0 ) a c>b c(c>0), a c<b c \quad(c<0) ac>bc(c>0),ac<bc(c<0)

a c > b c ( c > 0 ) , a c < b c ( c < 0 ) \frac{a}{c}>\frac{b}{c}(c>0), \frac{a}{c}<\frac{b}{c}(c<0) ca​>cb​(c>0),ca​<cb​(c<0)

a n > b n ( n > 0 , a > 0 , b > 0 ) , a n < b n ( n < 0 , a > 0 , b > 0 ) a^{n}>b^{n}(n>0, a>0, b>0), a^{n}<b^{n}(n<0, a>0, b>0) an>bn(n>0,a>0,b>0),an<bn(n<0,a>0,b>0)

a n > b n ( n  为正整数  , a > 0 , b > 0 ) \sqrt[n]{a}>\sqrt[n]{b} \quad(n \text { 为正整数 }, a>0, b>0) na ​>nb ​(n 为正整数 ,a>0,b>0)

对于任意实数 a , b a,b a,b,均有

∣ ∣ a ∣ − ∣ b ∣ ∣ ≤ ∣ a + b ∣ ≤ ∣ a ∣ + ∣ b ∣ || a|-| b|| \leq|a+b| \leq|a|+|b| ∣∣a∣−∣b∣∣≤∣a+b∣≤∣a∣+∣b∣

a 2 + b 2 ≥ 2 a b a^{2}+b^{2} \geq 2 a b a2+b2≥2ab

7 常用数列公式

等差数列 a 1 , a 1 + d , a 1 + 2 d , ⋯   , a 1 + ( n − 1 ) d a_{1}, a_{1}+d, a_{1}+2 d, \cdots, a_{1}+(n-1) d a1​,a1​+d,a1​+2d,⋯,a1​+(n−1)d,其公差为 n n n,前 n n n项的和为

s n = a 1 + ( a 1 + d ) + ( a 1 + 2 d ) + ⋯ + [ a 1 + ( n − 1 ) d ] = a 1 + [ a 1 + ( n − 1 ) d ] 2 ⋅ n s_{n}=a_{1}+\left(a_{1}+d\right)+\left(a_{1}+2 d\right)+\cdots+\left[a_{1}+(n-1) d\right]=\frac{a_{1}+\left[a_{1}+(n-1) d\right]}{2} \cdot n sn​=a1​+(a1​+d)+(a1​+2d)+⋯+[a1​+(n−1)d]=2a1​+[a1​+(n−1)d]​⋅n

( 首 项 + 尾 项 ) × 项 数 2 \frac{(首项+尾项)\times 项数}{2} 2(首项+尾项)×项数​

等比数列 a 1 , a 1 q , a 1 q 2 , ⋯   , a 1 q n − 1 , 公 比 为 a_{1}, a_{1} q, a_{1} q^{2}, \cdots, a_{1} q^{n-1},公比为 a1​,a1​q,a1​q2,⋯,a1​qn−1,公比为 q q q,前 n n n项的和为

s n = a 1 + a 1 q + a 1 q 2 + ⋯ + a 1 q n − 1 = a 1 ( 1 − q n ) 1 − q s_{n}=a_{1}+a_{1} q+a_{1} q^{2}+\cdots+a_{1} q^{n-1}=\frac{a_{1}\left(1-q^{n}\right)}{1-q} sn​=a1​+a1​q+a1​q2+⋯+a1​qn−1=1−qa1​(1−qn)​

一些常见数列的前 n n n项和

1 + 2 + 3 + ⋯ + n = 1 2 n ( n + 1 ) 1+2+3+\cdots+n=\frac{1}{2} n(n+1) 1+2+3+⋯+n=21​n(n+1)

2 + 4 + 6 + ⋯ + 2 n = n ( n + 1 ) 2+4+6+\cdots+2 n=n(n+1) 2+4+6+⋯+2n=n(n+1)

1 + 3 + 5 + ⋯ + ( 2 n − 1 ) = n 2 1+3+5+\cdots+(2 n-1)=n^{2} 1+3+5+⋯+(2n−1)=n2

1 2 + 2 2 + 3 2 + ⋯ + n 2 = 1 6 n ( n + 1 ) ( 2 n + 1 ) 1^{2}+2^{2}+3^{2}+\cdots+n^{2}=\frac{1}{6} n(n+1)(2 n+1) 12+22+32+⋯+n2=61​n(n+1)(2n+1)

1 2 + 3 2 + 5 2 + ⋯ + ( 2 n − 1 ) 2 = 1 3 n ( 4 n 2 − 1 ) 1^{2}+3^{2}+5^{2}+\cdots+(2 n-1)^{2}=\frac{1}{3} n\left(4 n^{2}-1\right) 12+32+52+⋯+(2n−1)2=31​n(4n2−1)

1 ⋅ 2 + 2 ⋅ 3 + 3 ⋅ 4 + ⋯ + n ( n + 1 ) = 1 3 n ( n + 1 ) ( n + 2 ) 1 \cdot 2+2 \cdot 3+3 \cdot 4+\cdots+n(n+1)=\frac{1}{3} n(n+1)(n+2) 1⋅2+2⋅3+3⋅4+⋯+n(n+1)=31​n(n+1)(n+2)

1 1 ⋅ 2 + 1 2 ⋅ 3 + 1 3 ⋅ 4 + ⋯ + 1 n ( n + 1 ) = 1 − 1 n + 1 \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{n(n+1)}=1-\frac{1}{n+1} 1⋅21​+2⋅31​+3⋅41​+⋯+n(n+1)1​=1−n+11​

1 3 + 2 3 + ⋯ + n 3 = ( 1 + 2 + ⋯ + n ) 2 1^{3}+2^{3}+\cdots+n^{3}=(1+2+\cdots+n)^{2} 13+23+⋯+n3=(1+2+⋯+n)2,即 ∑ i = 1 n i 3 = ( ∑ i = 1 n i ) 2 = [ n ( n + 1 ) 2 ] 2 \sum_{i=1}^{n} i^{3}=\left(\sum_{i=1}^{n} i\right)^{2}=\left[\frac{n(n+1)}{2}\right]^{2} ∑i=1n​i3=(∑i=1n​i)2=[2n(n+1)​]2

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