\(\left(20+9\frac{1}{4}\right):2\frac{1}{4}\) \(\left(6-2\frac{4}{5}\right).3\frac{1}{8}-1\frac{3}{5}:\frac{1}{4}\)

\(\frac{32}{15}:\left(-1\frac{1}{5}+1\frac{1}{3}\right)\) \(0,2.\frac{15}{36}-\left(\frac{2}{5}=\frac{2}{3}\right):1\frac{1}{5}\)

\(\frac{-3}{7}.\frac{5}{9}+\frac{4}{9}.\frac{-3}{7}+2\frac{3}{7}\) \(0,7.2\frac{2}{3}.20.0,375.\frac{5}{8}\)

1. Cho I =\(\frac{1}{4^2}+\)\(\frac{1}{6^2}+\)\(\frac{1}{8^2}+...+\frac{1}{2006^2}\)chứng minh: I < \(\frac{334}{2007}\)
2. Cho K = \(\frac{1}{5^2}+\frac{1}{9^2}+...+\frac{1}{409^2}\)chứng minh: K < \(\frac{1}{12}\)
3. cho M = \(\frac{3}{4}+\frac{8}{9}+\frac{15}{16}+...+\frac{2499}{2500}\)chứng minh: M > 48
4. cho N = \(\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\frac{1}{1+2+3+4+5}+...+\frac{1}{1+2+3+4+...+59}\)chứng minh: N < \(\frac{2}{3}\)
5. cho S = \(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{37.38.39}\)chứng minh: S <\(\frac{1}{4}\)

 Chứng minh rằng :

a) S=\(\frac{3}{1.4}\)+\(\frac{3}{4.7}\)+...+\(\frac{3}{n(n+3)}\)< 1 .

b) Cho : S=\(\frac{1}{2^2}\)+\(\frac{1}{3^2}\)+\(\frac{1}{4^2}\)+...+\(\frac{1}{9^2}\). Chứng minh :\(\frac{2}{5}\)< S <\(\frac{8}{9}\).

   Gợi ý : Chứng minh \(\frac{1}{b}\)-\(\frac{1}{b+1}\)<\(\frac{1}{b^2}\)<\(\frac{1}{b-1}\)-\(\frac{1}{b}\) ( b\(\varepsilon\)\(ℕ\), b > 1 ) .

Bài 1:

a, Cho S=\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{9^2}\) .Chứng minh rằng \(\frac{2}{5}< S< \frac{8}{9}\)

b, Tìm x thuộc z để phân số \(\frac{x^2-5x-1}{x+2}\)có giá trị là số nguyên

c, Chứng minh rằng \(\left(\frac{7}{65}+1\right)\left(\frac{7}{84}+1\right)\left(\frac{7}{105}+1\right)\left(\frac{7}{124}+1\right)...\left(\frac{7}{153+1}\right)\left(\frac{7}{560}+1\right)< 2\)

d, Chứng minh rằng \(\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}-\frac{4}{3^4}+\frac{5}{3^5}-...+\frac{99}{3^{99}}-\frac{100}{3^{100}}< \frac{3}{16}\)

BÀI 1:

a/ \(\frac{2}{7}+\frac{-3}{8}+\frac{11}{7}+\frac{1}{3}+\frac{1}{7}+\frac{5}{-8}\)

b/ \(\frac{-3}{8}+\frac{12}{25}+\frac{5}{-8}+\frac{2}{-5}+\frac{13}{25}\)

c/ \(\frac{7}{8}+\frac{1}{8}.\frac{3}{8}+\frac{1}{8}.\frac{5}{8}\)

d/ \(\frac{-5}{6}.\frac{4}{19}+\frac{-7}{12}.\frac{4}{19}-\frac{40}{57}\)

e/ \(\frac{3}{7}.\frac{9}{26}-\frac{1}{14}.\frac{1}{13}-\frac{1}{7}\)

f/ \(\left(\frac{2}{3}-\frac{1}{ }_{_4+\frac{5}{11}}\right):\left(\frac{5}{12}+1-\frac{7}{11}\right)\)

g/ \(\frac{4}{9}:\left(-\frac{1}{7}\right)+6\frac{5}{9}:\left(-\frac{1}{7}\right)\)

h/ \(1\frac{5}{18}-\frac{5}{18}:\left(\frac{1}{15}+1\frac{1}{12}\right)\)

i/ \(\frac{-1}{7}.\left(9\frac{1}{2}-8,75\right):\frac{2}{7}+62,5\%:1\frac{2}{3}\)

bài 1 thực hiện phép tính, tính nhanh nếu có thể

a, \(\frac{-11}{23}.\frac{6}{7}+\frac{8}{7}.\frac{-11}{23}-\frac{1}{23}\)

b,\(0,75.1\frac{7}{9}-1\frac{2}{5}:\frac{-21}{20}\)

c,\(2.\frac{3}{7}+\left(\frac{2}{9}-1\frac{3}{7}\right)-\frac{5}{3}:\frac{1}{9}\)

d,\(11\frac{1}{4}-\left(2\frac{5}{7}+5\frac{1}{4}\right)\)

e,\(\left(6-2\frac{4}{5}\right).3\frac{1}{8}-1\frac{3}{5}:25\%\)

g,\(0,2.\frac{15}{36}-\left(\frac{2}{5}+\frac{2}{3}\right):1\frac{1}{5}\)

h,\(2\frac{1}{3}-\frac{1}{3}.\left(\frac{-3}{2}+\left(\frac{2}{3}+0,4.5\right)\right)\)

i,\(\frac{1}{3}.\frac{5}{7}-\frac{7}{27}.\frac{36}{14}\)

k,\(\left(\frac{-5}{28}+1,75+\frac{8}{35}\right):\left(-3\frac{9}{20}\right)\)

l,\(\left(20+9\frac{1}{4}\right):2\frac{1}{4}\)

m,\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{12.13}\)

n,\(\frac{5}{1.3}+\frac{5}{3.5}+\frac{5}{5.7}+...+\frac{5}{99.101}\)

Bài 1:Chứng tỏ rằng

a)\(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{2009.2010}< 1\)

b)\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}< 1\)

c)\(\frac{2}{5}< \frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{9^2}< \frac{8}{9}\)

d)\(\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}-\frac{4}{3^4}+...+\frac{99}{3^{99}}-\frac{100}{3^{100}}< \frac{3}{16}\)

Bài 2:Cho M=\(\frac{1}{15}+\frac{1}{105}+\frac{1}{315}+..+\frac{1}{9177}\).So sánh với 12

Bài 3:Với giá trị nào của x \(\in\) Z các phân số sau có giá trị là 1 số nguyên

a)A=\(\frac{3}{x-1}\) b)B=\(\frac{x-2}{x+3}\) c)C=\(\frac{2x+1}{x-3}\) d)D=\(\frac{x^2-1}{x+1}\)

Bài 4:a) Chứng tỏ rằng các phân số sau tối giản với mọi số tự nhiên n

a)\(\frac{n+1}{2n+3}\) b)\(\frac{2n+3}{4n+8}\)

Mình đang cần gấp lắm ,làm ơn

Bài 5 : Chững minh rẳng :
a) S= \(\frac{3}{10}+\frac{3}{11}+\frac{3}{12}+\frac{3}{13}+\frac{3}{14}\) CMR :1< S <2
b) \(\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{8^2}+\frac{1}{10^2}+...+\frac{1}{\left(2n\right)^2}< \frac{1}{4}\)
c) \(\frac{3}{4}+\frac{8}{9}+\frac{15}{16}+...+\frac{2499}{2500}>48\)
d) \(\frac{1}{5}+\frac{1}{13}+\frac{1}{14}+\frac{1}{15}+\frac{1}{61}+\frac{1}{62}+\frac{1}{63}< \frac{1}{2}\)
Bài :So sánh phân số sau:
a)\(\frac{1985.1987-1}{1980+1985.1986}và1\)
b) A= \(\frac{13^{15}+1}{13^{16}+1}\)và B = \(\frac{13^{16}+1}{13^{17}+1}\)
c)\(\frac{18}{53}và\frac{26}{79}\)
d)\(\frac{5}{8}và\frac{14}{17}\)
e)\(\frac{1}{5^{199}}và\frac{1}{3^{300}}\)
g)\(\frac{1}{3^{17}}và\frac{1}{5^{10}}\)
h) \(\frac{18}{109}và\frac{5}{30}\)