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\(M=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{10.11}-\frac{1}{11.12}\)
\(=\frac{1}{2}-\frac{1}{11.12}\)
\(=\frac{65}{132}\)
M=1/2{1/3-1/5+1/5-1/7+1/7-1/9+...+1/97-1/99}
=1/2{1/3-1/99}
=1/2*32/99
=16/99
Đặt \(A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{18.19.20}\)
\(\Rightarrow2A=\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{18.19.20}\)
\(=\left(\frac{1}{1.2}-\frac{1}{2.3}\right)+\left(\frac{1}{2.3}-\frac{1}{3.4}\right)+...+\left(\frac{1}{18.19}-\frac{1}{19.20}\right)\)
\(=\frac{1}{2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{18.19}-\frac{1}{19.20}\)
\(=\frac{1}{2}-\frac{1}{19.20}<\)\(\frac{1}{2}\)
\(2A<\)\(\frac{1}{2}\)
\(\Rightarrow A<\)\(\frac{1}{4}\)
Vậy \(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{18.19.20}<\)\(\frac{1}{4}\)
\(\frac{3}{4}\)-\(\frac{-5}{9}\)-\(\frac{11}{36}\)=\(\frac{27}{36}\)-\(\frac{-20}{36}\)-\(\frac{11}{36}\)=1
\(\frac{1}{9}\)+\(\frac{-5}{3}\)-\(\frac{-13}{18}\)=\(\frac{2}{18}\)+\(\frac{-30}{18}\)-\(\frac{-13}{18}\)=\(\frac{-15}{18}\)=\(\frac{-5}{6}\)
\(\frac{3}{4}-\frac{-5}{9}-\frac{11}{36}=\frac{27}{36}-\frac{-20}{36}-\frac{11}{36}=\frac{47}{36}-\frac{11}{36}=\frac{36}{36}=1\)
\(\frac{1}{9}+\frac{-5}{3}-\frac{13}{18}=\frac{2}{18}+\frac{-30}{18}-\frac{13}{18}=\frac{-28}{18}+\frac{13}{18}=\frac{-15}{18}=\frac{-5}{6}\)
M=1+1/2^2+1/3^2+1/4^2+...+1/10^2>1+1/2*3+1/3*4+1/4^5+...+1/10*11
M>1+1/2-1/3+1/4-1/4+1/5-...-1/11
M>1+1/2-1/11
M>1+9/22
M>31/22
vì 31/22>4/3 nên M>4/3
Câu 1 :\(P=\left(1-\frac{1}{2}\right).\left(1-\frac{1}{3}\right).....\left(1-\frac{1}{99}\right)=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}.....\frac{98}{100}=\frac{1}{100}\)
gọi A=1/1*2*3+1/2*3*4+...+1/49*50*51
2A=2(1/1*2*3+1/2*3*4+...+1/49*50*51)
2A=2/1*2*3+2/2*3*4+...+2/49*50*51
2A=1/1*2-1/2*3+1/2*3-1/3*4+...+1/49*50-1/50*51
2A=1/2-1/2550
2A=637/1275
A=637/1275:2
A=637/2550
qua bài trên ta có công thức \(\frac{1}{n\cdot\left(n+1\right)\cdot\left(n+2\right)}\)= \(\frac{1}{n\cdot\left(n+1\right)}\)-\(\frac{1}{\left(n+1\right)\cdot\left(n+2\right)}\)
lộn công thức là 2/n*(n+1)*(n+2)=1/n*(n+1)-1/(n+1)*(n+2) cho tui xin lỗi
mà tick nhé