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\(A< \frac{1}{2^2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2018.2019}=\frac{1}{2^2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2018}-\frac{1}{2019}\)
=> \(A< \frac{1}{2^2}+\frac{1}{2}-\frac{1}{2019}=\frac{3}{4}-\frac{1}{2019}=\frac{3}{4}\)
Vậy A<3/4
A< \(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+.....+\frac{1}{2018.2019}\)
=\(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2018}-\frac{1}{2019}\)
=\(1-\frac{1}{2019}=\frac{2019-1}{2019}=\frac{2018}{2019}\)
\(P=\frac{1}{5^2}+\frac{2}{5^3}+\frac{3}{5^4}+\frac{4}{5^5}+...+\frac{11}{5^{12}}\)
\(\Rightarrow\)\(5P=\frac{1}{5}+\frac{2}{5^2}+\frac{3}{5^3}+\frac{4}{5^4}+...+\frac{11}{5^{11}}\)
\(\Rightarrow\)\(4P=\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+\frac{1}{5^4}+...+\frac{1}{5^{11}}-\frac{1}{5^{12}}\)
\(\Rightarrow\)\(20P=1+\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+...+\frac{1}{5^{10}}-\frac{1}{5^{11}}\)
\(\Rightarrow\)\(16P=1-\frac{1}{5^{11}}+\frac{1}{5^{12}}-\frac{1}{5^{11}}\)\(< 1\)
\(\Rightarrow\)\(P< \frac{1}{16}\)
P/s: nguyên tác: https://olm.vn/thanhvien/nhatphuonghocgiot
\(1)A=a\frac{1}{3}+a\frac{1}{4}-a\frac{1}{6}=a\left(\frac{1}{3}+\frac{1}{4}-\frac{1}{6}\right)=a\frac{5}{12}\)
Thay \(a=-\frac{3}{5}\) vào A,ta đc:
\(A=-\frac{3}{5}.\frac{5}{12}=-\frac{1}{4}\)
\(2)B=b\frac{5}{6}+b\frac{3}{4}-b\frac{1}{2}=b\left(\frac{5}{6}+\frac{3}{4}-\frac{1}{2}\right)=b\frac{13}{12}\)
Thay \(b=\frac{12}{13}\) vào B, ta đc: \(B=b\frac{13}{12}=\frac{12}{13}.\frac{13}{12}=1\)
S= - 32\(\left(\frac{1}{4}+\frac{1}{28}+\frac{1}{70}+...+\frac{1}{868}\right)\)
S = - 32\(\left(\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+...+\frac{1}{28.31}\right)\)
S = - 3\(\left(\frac{3}{1.4}+\frac{3}{4.7}+...+\frac{3}{28.31}\right)\)
S = -3\(\left(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+...+\frac{1}{28}-\frac{1}{31}\right)\)
S = -3 \(\left(1-\frac{1}{31}\right)\)
S = -3\(.\frac{30}{31}\)
S = -90/31
1/3S=-(1/1*4+1/4*7+1/7*10+...+1/28*31)=-(1/1-1/4+1/4-1/7+1/7-1/10+...+1/28-1/31)=-(1/1-1/31)=-30/31
=>S=(-30/31):1/3=-90/31
Có \(\frac{2!}{3!}=\frac{1}{3}>\frac{1}{4}\)nên nó lớn hơn 1/4
kick minh di mink giai cho de lam