Bài 1:Tìm x, biết: \(\left(x-1\right)^{x+2}=\left(x-1\right)^{x+4}\)

Bài 2:Tính giá trị của :

\(M=100^2-99^2+98^2-97^2+...+2^2-1^2\)

\(N=\left(20^2+18^2+16^2+...+4^2+2^2\right)\)\(-\left(19^2+17^2+15^2+...+3^2+1^2\right)\)

\(P=\left(-1\right)^n.\left(-1\right)^{2n+1}.\left(-1\right)^{n+1}\)

Giúp mình với mình cần gấp

1. Tính

a)A=\(\left(1-\frac{1}{1+2}\right)\left(1-\frac{1}{1+2+3}\right)...\left(1-\frac{1}{1+2+3+...+2006}\right)\)

b)\(B=1+\frac{3}{2^3}+\frac{4}{2^4}+...+\frac{100}{2^{100}}\)

c)C=\(\frac{1}{2!}+\frac{2}{3!}+...+\frac{n-1}{n!}\)

d)D=\(1+2^2+3^2+...+98^2\)

e)E=\(3^{100}-3^{99}+3^{98}-3^{97}+...+3^2-3+1\)

f)F=\(2^{2010}-2^{2009}-...-2-1\)

g)G=\(\left(\frac{1}{4}-1\right)\left(\frac{1}{9}-1\right)\left(\frac{1}{16}-1\right)...\left(\frac{1}{100}-1\right)\left(\frac{1}{121}-1\right)\)

Tính :

1 + 3 + 5 + 7 + ... + (2n - 1) = 225 

Giải :

Theo công thức tính dãy số , ta có :

\(\frac{\left\{\left[\left(2n-1\right)-1\right]:2+1\right\}.\left[\left(2n-1\right)+1\right]}{2}=225\)

\(\frac{\left\{\left[2n-2\right]:2+1\right\}.2n}{2}=225\)

\(\left\{\left[2n-2\right]:2+1\right\}.n=450\)(Lượt giản thừa số 2)

\(\left\{\frac{2n-2}{2}+1\right\}.n=225\)

\(\left\{\frac{2n-2}{2}+\frac{2}{2}\right\}.n=225\)

\(\frac{2n-2+2}{2}.n=225\)

\(\frac{2n}{2}.n=225\)

\(n^2=225\)

\(\Rightarrow n=\sqrt{225}=15\)

Bài 1: Tính

a. \(\left(1+\frac{1}{1\cdot3}\right)\cdot\left(1+\frac{1}{2\cdot4}\right)\cdot\left(1+\frac{1}{3\cdot5}\right)+\left(1+\frac{1}{4\cdot6}\right).....\left(1+\frac{1}{99\cdot101}\right)\)

b. \(\left[\sqrt{0,64}+\sqrt{0,0001}-\sqrt{\left(-0,5\right)^2}\right]\div\left[3\cdot\sqrt{\left(0,04\right)^2}-\sqrt{\left(-2\right)^4}\right]\)

c. \(\frac{5.4^{15}\cdot9^9-4.3^{20}\cdot8^9}{5\cdot2^9\cdot6^{19}-7\cdot2^{29}\cdot27^6}-\frac{2^{19}\cdot6^{15}-7\cdot6^{10}\cdot2^{20}\cdot3^6}{9\cdot6^{19}\cdot2^9-4\cdot3^{17}\cdot2^{26}}+0,\left(6\right)\)

Bài 2: Tìm x, y, z biết :
a. \(\left(x-10\right)^{1+x}=\left(x-10\right)^{x+2009}\left(x\in Z\right)\)

b. \(\left|x-2007\right|+\left|x-2008\right|+\left|y-2009\right|+\left|x-2010\right|=3\left(x,y\in N\right)\) 

c. \(25-y^2=8\left(x-2009\right)^2\left(x,y\in Z\right)\)

d. \(2008\left(x-4\right)^2+2009\left|x^2-16\right|+\left(y+1\right)^2\le0\)

e. \(2x=3y\) ; \(4z=5x\) và \(3y^2-z^2=-33\)

Bài 3: Chứng minh rằng

a. \(1-\frac{1}{2^2}-\frac{1}{3^2}-\frac{1}{4^2}-...-\frac{1}{2009^2}>\frac{1}{2009}\)

b. \(\left[75\cdot\left(4^{2008}+4^{2007}+4^{2006}+...+4+1\right)+25\right]⋮100\)

Bài 4: 

a. Tìm giá trị nhỏ nhất của biểu thức : \(M=\left(x^2+2\right)+\left|x+y-2009\right|+2005\)

b. So sánh: \(31^{11}\) và \(\left(-17\right)^{14}\)

c. So sánh: \(\left(\frac{9}{11}-0,81\right)^{2012}\) và \(\frac{1}{10^{4024}}\)