Với n thuộc N* chứng minh
\(\frac{1}{n\left(n+1\right)\left(n+2\right)}=\frac{1}{2}\left[\frac{1}{n\left(n+1\right)}-\frac{1}{\left(n+1\right)\left(n+2\right)}\right]\)
Bài 1:
1. Tính: \(E=1+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+\frac{1}{4}\left(1+2+3+4\right)+...+\frac{1}{200}\left(1+2+...+200\right)\)
2. Tìm và tính tổng các số nguyên x thỏa mãn: \(\frac{21}{5}\left|x\right|< 2019\)
3. Tìm x, biết: \(\frac{2^{24}\left(x-3\right)}{\left(3\frac{5}{7}-1,4\right)\left(6\cdot2^{24}-4^{13}\right)}=\left(\frac{5}{3}\right)^2\)
1 TÍNH
\(a,\left(\frac{-1}{4}\right)^0\)
\(b,\left(-2\frac{1}{3}\right)^2\)
\(c,\left(\frac{4}{5}\right)^{-2}\)
\(d,\left(0,5\right)^{-3}\)
\(e,\left(-1\frac{1}{3}\right)^4\)
\(f,27^3:3^2\)
\(g,\left(\frac{3}{5}\right)^{15}:\left(\frac{9}{25}\right)^5\)
\(h,5-\left(-\frac{5}{11}\right)^0+\left(\frac{1}{3}\right)^2:3\)
\(i,\left(\frac{1}{3}\right)^{-3}+3.\left(\frac{1}{2}\right)^0+\left[\left(-2\right)^2:\frac{1}{2}\right].8\)
Cho \(B=1+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+\frac{1}{4}\left(1+2+3+4\right)+...+\frac{1}{x}\left(1+2+3+4+...+x\right)\)
Tìm số nguyên dương x để B= 115
4,Tìm x biết
a,\(\frac{-2}{3}\cdot x+\frac{1}{5}=\frac{3}{10}\)
b,\(\frac{2}{3}\cdot x-\frac{3}{2}\cdot x=\frac{5}{12}\)
c,\(\left(4,5-2x\right)\cdot\left(-1\frac{4}{7}\right)=\frac{11}{14}\)
d,\(\frac{1}{4}+\frac{1}{3}:3x=-5\)
e,\(\left(2\frac{4}{5}x-50\right):\frac{2}{3}=51\)
g,\(|4x-1|=\left(-3^2\right)\)
h,\(|x+70|=2\frac{1}{5}\)
i,\(\left(x-1^3\right)=125\)
k,\(\left(x+\frac{1}{2}\right)\cdot\left(\frac{2}{3}-2x\right)=0\)
Giúp mình nhé!!!
Bài 1: Tính a) \(\left(\frac{11}{12}:\frac{44}{16}\right)\cdot\left(\frac{-1}{3}+\frac{1}{2}\right)\) b) \(\frac{\left(-5^2\right)\cdot\left(-5\right)^3\cdot16}{5^4\cdot\left(-2\right)^4}\) c) \(7,5:\left(\frac{-5}{3}\right)+2\frac{1}{2}:\left(\frac{-5}{3}\right)\)d) \(\left(\frac{-1}{2}+\frac{1}{3}\right)\cdot\frac{4}{5}+\left(\frac{2}{3}+\frac{1}{2}\right):\frac{4}{5}\)
Bài 1: Cho \(A=\frac{196}{197}+\frac{197}{198}\) và \(B=\frac{196+197}{197+198}\). Trong 2 số A và B, số nào lớn hơn?
Bài 2: Tính nhanh: \(1+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+\frac{1}{4}\left(1+2+3+4\right)+...+\frac{1}{16}\left(1+2+3+...+16\right)\)
Tính giá trị của :
D=\(\left(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2019^2}\right)x\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2020^2}\right)-\left(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2020^2}\right)x\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2019^2}\right)\)
Bài 1: Chứng minh rằng: \(\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}-\frac{4}{3^4}+...+\frac{99}{3^{99}}-\frac{100}{3^{100}}< \frac{3}{16}\)
Bài 2: Cho \(n\in N;n>1\). Chứng minh rằng: \(S=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{\left(n-1\right)^2}+\frac{1}{n^2}\notin N\)