a,A=\(\frac{1}{2}-\frac{1}{2^2}+\frac{1}{2^3}-\frac{1}{2^4}+...+\frac{1}{2^{99}}-\frac{1}{2^{100}}\)
b,B=\(\frac{1}{1\times2\times3}+\frac{1}{2\times3\times4}+\frac{1}{3\times4\times5}+...+\frac{1}{998\times999\times100}\)
c,C=\(\frac{1+\left(1+2\right)+\left(1+2+3\right)+...+\left(1+2+3+...+98\right)}{1\times98+2\times97+3\times96+...+98\times1}\)
CMR:Với mọi số tự nhiên n \(\ne\)0 ta đều có:
a.\(\frac{1}{2\times5}+\frac{1}{5\times8}+\frac{1}{8\times11}+...+\frac{1}{\left(3n-1\right)\times\left(3n+2\right)}=\frac{1}{6n+4}\)
b.\(\frac{5}{3\times7}+\frac{5}{7\times11}+\frac{5}{11\times15}+...+\frac{5}{\left(4n-1\right)\times\left(4n+3\right)}=\frac{5n}{4n+3}\)
Tính nhanh giá trị biểu thức sau:
a) \(-\frac{9}{10}\cdot\frac{5}{14}+\frac{1}{10}\cdot\left(-\frac{9}{2}\right)+\frac{1}{7}\cdot\left(-\frac{9}{10}\right)\)
b)\(\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}+\frac{1}{11}\right)\cdot132\)
c)\(-\frac{2}{3}\cdot\left(\frac{8}{9}\cdot\frac{8}{13}-\frac{8}{27}\cdot\frac{3}{13}+\frac{4}{3}\cdot\frac{22}{39}\right)\)
a,Cho A=\(\left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{98}\right)\cdot2\cdot3\cdot4\cdot...\cdot98\)
CMR:A chia hết cho 99
b,Cho B=\(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{96}\) và B bằng phân số \(\frac{a}{b}\) .CMR A chia hết cho 97
tính
B=\(\frac{\left(\frac{1}{2}\right)^{10}.5-\left(\frac{1}{4}\right)^5.3}{\frac{1}{1024}.\frac{1}{3}-\left(\frac{1}{2}\right)^{11}}\)
bài 1 : tính
a)\(\frac{-5}{13}-\left(\frac{3}{5}+\frac{3}{13}-\frac{4}{10}\right)\) b) \(\left(\frac{3}{9}-\frac{9}{18}\right)+\frac{3}{6}-\left(\frac{1}{3}-\frac{1}{2}\right)-\frac{5}{15}\) c) \(\frac{9}{18}+\frac{16}{32}-\frac{12}{46}-\frac{9}{17}\) d) \(\left(\frac{14}{18}+\frac{-16}{27}\right)-\left(\frac{2}{3}-\frac{5}{15}\right)\)
CMR:Với mọi số tự nhiên n \(\ne\)0 ta đều có:
a.\(\frac{1}{2\cdot5}+\frac{1}{5\cdot8}+\frac{1}{8\cdot11}+...+\frac{1}{\left(3n-1\right)\cdot\left(3n+2\right)}=\frac{n}{6n+4}\)
b.\(\frac{5}{3\cdot7}+\frac{5}{7\cdot11}+\frac{5}{11\cdot15}+...+\frac{5}{\left(4n-1\right)\cdot\left(4n+3\right)}=\frac{5n}{4n+3}\)
Tìm x biết:
a) \(^{2^x+2^{x+1}+2^{x+2}+2^{x+3}=480}\)
b) \(\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2012}+\frac{1}{2013}\right).x=\frac{2012}{1}+\frac{2011}{2}+\frac{2010}{3}+...+\frac{2}{2011}+\frac{1}{2012}\)
Cho B=\(\left(1-\frac{1}{2}\right)\cdot\left(1-\frac{1}{3}\right)\cdot\left(1-\frac{1}{4}\right)\cdot...\cdot\left(1-\frac{1}{20}\right)\)
So sánh B với \(\frac{1}{21}\)
Tìm x,y biết :
a) \(\frac{3}{4}x+\frac{1}{5}=-\frac{1}{2}\)
b)\(-\frac{4}{5}+\left(\frac{1}{3}:y\right)=-\frac{5}{6}\)