Cho \(A=\dfrac{1}{11}+\dfrac{1}{12}+\dfrac{1}{13}+...+\dfrac{1}{70}\)
Chứng minh rằng : a) \(A>\dfrac{4}{3}\)
b) \(A< 2,5\)
Chứng minh rằng :
a) \(\dfrac{1}{2!}+\dfrac{1}{3!}+\dfrac{1}{4!}+...+\dfrac{1}{100!}< 1\)
b) \(\dfrac{9}{10!}+\dfrac{9}{11!}+\dfrac{9}{12!}+...+\dfrac{9}{1000!}< \dfrac{1}{9!}\)
Cho A= \(\dfrac{1}{5^2}+\dfrac{2}{5^3}+\dfrac{3}{5^4}+...+\dfrac{11}{5^{12}}\) với n\(\in N\)
Chứng minh rằng a < \(\dfrac{1}{16}\)
a) (0,75 - 1/4): 5/6
b) (3/4-0,25):3/7
c)1 \(\dfrac{13}{15}\).(0.5)^2 .3+(8/15- 1\(\dfrac{19}{60}\)):1\(\dfrac{23}{24}\)
d) (3\(\dfrac{1}{3}\)+2,5):(3\(\dfrac{1}{6}\)-4\(\dfrac{1}{5}\))-\(\dfrac{11}{31}\)
e) [6+(1/2)^3- /-\(\dfrac{1}{2}\)/ ] :3/12
Tính nhanh:
1) \(\dfrac{2}{15}.6\dfrac{5}{11}+\dfrac{5}{11}.\dfrac{-2}{15}-\dfrac{2}{15}.2015^0\)
2)\(\dfrac{5}{2.7}+\dfrac{3}{14.11}+ \dfrac{4}{11.7}+\dfrac{1}{14.15}+\dfrac{13}{15.16}\)
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Chứng minh rằng các tổng sau không phải là số tự nhiên :
a) \(A=\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}\)
b) \(B=\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{8}\)
c) \(C=\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{16}\)
BT2: Tính nhanh
11) \(-\dfrac{5}{7}-\left(-\dfrac{5}{67}\right)+\dfrac{13}{30}+\dfrac{1}{2}+\left(-1\dfrac{5}{6}\right)+1\dfrac{3}{14}-\left(-\dfrac{2}{5}\right)\)
12) \(\dfrac{-1}{4}.13\dfrac{9}{11}-0.25.6\dfrac{2}{11}\)
bài 4 chứng minh
a) \(\dfrac{11}{15}< \dfrac{1}{21}+\dfrac{1}{22}+...+\dfrac{1}{60}< \dfrac{3}{2}\)
b) \(3< 1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{63}< 6\)
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Bài 1: Thực hiện phép tính
a) \(\left(\dfrac{3}{8}+\dfrac{-3}{4}+\dfrac{7}{12}\right):\dfrac{5}{6}+\dfrac{1}{2}\)
b) \(\dfrac{1}{2}+\dfrac{3}{4}-\left(\dfrac{3}{4}-\dfrac{4}{5}\right)\)
c) \(6\dfrac{5}{12}:2\dfrac{3}{4}+11\dfrac{1}{4}.\left(\dfrac{1}{3}\dfrac{1}{5}\right)\)
d)\(\left(\dfrac{7}{8}-\dfrac{3}{4}\right).1\dfrac{1}{3}-\dfrac{2}{7}.\left(3,5\right)^2\)
e)\(\left(\dfrac{3}{5}+0,415-\dfrac{3}{200}\right).2\dfrac{2}{3}.0,25\)
f)\(\dfrac{5}{16}:0,125-\left(2\dfrac{1}{4}-0,6\right).\dfrac{10}{11}\)
g) \(0,25:\left(10,3-9,8\right)-\dfrac{3}{4}\)
h)\(1\dfrac{13}{15}.0,75-\left(\dfrac{11}{20}+20\%\right):\dfrac{7}{3}\)
i)\(\dfrac{\left(\dfrac{1}{2-0,75}\right).\left(0,2-\dfrac{2}{5}\right)}{\dfrac{5}{9}-1\dfrac{1}{12}}\)
k)\(\dfrac{\dfrac{2}{3}+\dfrac{2}{7}-\dfrac{1}{14}}{-1-\dfrac{3}{7}+\dfrac{3}{28}}\)
Bài 2: Tính bằng cánh hợp lí
A=\(0,7.2\dfrac{2}{3}.20.0,375.\dfrac{5}{28}\)
B=\(\left(9\dfrac{30303}{80808}+7\dfrac{303030}{484848}\right)+4,03\)
C=\(\left(9,75.21\dfrac{3}{7}+\dfrac{39}{4}.18\dfrac{4}{7}\right).\dfrac{15}{78}\)
D=\(10101.\left(\dfrac{5}{111111}+\dfrac{5}{222222}-\dfrac{4}{3.7.11.13.37}\right)\)