Ta có:
\(\dfrac{1}{a}-\dfrac{1}{a+1}\)
\(=\dfrac{a+1}{a\left(a+1\right)}-\dfrac{a}{a\left(a+1\right)}\)
\(=\dfrac{a+1-a}{a\left(a+1\right)}\)
\(=\dfrac{1}{a\left(a+1\right)}\left(Đpcm\right)\)
Cho \(A=\left(\dfrac{2}{2^2}-1\right).\left(\dfrac{1}{3^2}-1\right).\left(\dfrac{1}{4^2}-1\right)...\left(\dfrac{1}{100^2}-1\right)\)So sánh A với \(-\dfrac{1}{2}\)
a) 12,5.\(\left(-\dfrac{5}{7}\right)+1,5.\left(-\dfrac{5}{7}\right)\)
b) \(\left(-\dfrac{2}{5}-\dfrac{3}{7}\right):\dfrac{4}{5}+\left(-\dfrac{1}{5}+\dfrac{3}{7}\right);\dfrac{4}{5}\)
c) \(12.\left(-\dfrac{2}{3}\right)^2+\dfrac{4}{3}\)
d) \(1:\left(\dfrac{2}{3}-\dfrac{3}{4}\right)^2\)
e) \(15.\left(-\dfrac{2}{3}\right)^{^{ }2}-\dfrac{7}{3}\)
f) \(\dfrac{5^4.20^4}{25^5.4^5}\)
a) Chứng tỏ rằng nếu \(\dfrac{a}{c}< \dfrac{c}{d}\left(b>0,d>0\right)\) thì \(\dfrac{a}{b}< \dfrac{a+c}{b+d}< \dfrac{c}{d}\)
b) Hãy viết ba số hữu tỉ xen giữa \(-\dfrac{1}{3}\) và \(-\dfrac{1}{4}\)
Bài 6 ( SBT toán 7 tập 1 / trang 6)
a) Chứng tỏ rằng nếu \(\dfrac{a}{b}< \dfrac{c}{d}\left(b>0,d>0\right)\) thì \(\dfrac{a}{b}< \dfrac{a+c}{b+d}< \dfrac{c}{d}\).
Tìm x, biết:
a) \(\dfrac{-3}{7}+x=\dfrac{1}{3}\)
b) \(\dfrac{3}{4}+\dfrac{1}{4}:x=\dfrac{2}{5}\)
c) \(1\dfrac{1}{3}:0,8=\dfrac{2}{3}:0,1x\)
d) \(\left|x-3\right|=\dfrac{1}{2}\)
Thực hiện phép tính:
a) \(\left(\dfrac{9}{25}-2.18\right):\left(3\dfrac{4}{5}+0,2\right)\)
b) \(\dfrac{3}{8}.19\dfrac{1}{3}-\dfrac{3}{8}.33\dfrac{1}{3}\)
c) \(1\dfrac{4}{23}+\dfrac{5}{21}-\dfrac{4}{23}+0,5+\dfrac{16}{21}\)
d) \(\dfrac{21}{47}+\dfrac{9}{45}+\dfrac{26}{47}+\dfrac{4}{5}\)
e) \(\dfrac{15}{12}+\dfrac{5}{13}-\dfrac{3}{12}-\dfrac{18}{13}\)
\(\dfrac{2}{\left(x-1\right)\left(x-3\right)}\)+\(\dfrac{2}{\left(x-3\right)\left(x-8\right)}+\dfrac{12}{\left(x-8\right)\left(x-20\right)}-\dfrac{1}{x-20}=\dfrac{-3}{4}\)
So sánh:
a, \(\left(\dfrac{1}{24}\right)^9\)và \(\left(\dfrac{1}{83}\right)^{13}\)
c, \(\dfrac{1}{5^{199}}\)và\(\dfrac{1}{3^{300}}\)
\(\dfrac{4}{5}-\left(\dfrac{-2}{7}\right)-\dfrac{7}{10}\)
\(\left(6-\dfrac{2}{3}+\dfrac{1}{2}\right)-\left(5+\dfrac{5}{3}-\dfrac{3}{2}\right)-\left(3-\dfrac{7}{3}+\dfrac{5}{2}\right)\)