Bài 1 :
a) \(\dfrac{12}{15}< \dfrac{12}{14}< \dfrac{13}{14}\Rightarrow\dfrac{12}{15}< \dfrac{13}{14}\)
b) \(\dfrac{11}{12}< \dfrac{11+1984}{12+1984}=\dfrac{1995}{1996}\)
\(\Rightarrow\dfrac{11}{12}< \dfrac{1995}{1996}\)
c) \(\dfrac{499}{498}>\dfrac{499+1}{498+1}=\dfrac{500}{499}\)
\(\Rightarrow\dfrac{499}{498}>\dfrac{500}{499}\)
d) \(\dfrac{51}{80}< \dfrac{51}{79}< \dfrac{53}{79}\)
\(\Rightarrow\dfrac{51}{80}< \dfrac{53}{79}\)
Bài 2:
a) \(\dfrac{1}{9\times10}+\dfrac{1}{10\times11}+\dfrac{1}{11\times12}+...+\dfrac{1}{29\times30}\)
\(=\dfrac{1}{9}-\dfrac{1}{10}+\dfrac{1}{10}-\dfrac{1}{11}+...+\dfrac{1}{29}-\dfrac{1}{30}\)
\(=\dfrac{1}{9}-\dfrac{1}{30}\)
\(=\dfrac{10}{90}-\dfrac{3}{90}\)
\(=\dfrac{7}{90}\)
b) \(\dfrac{2}{7\times9}+\dfrac{2}{9\times11}+\dfrac{2}{11\times13}+....+\dfrac{2}{37\times39}\)
\(=\dfrac{1}{7}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{11}+...+\dfrac{1}{37}-\dfrac{1}{39}\)
\(=\dfrac{1}{7}-\dfrac{1}{39}\)
\(=\dfrac{32}{273}\)
c) \(\dfrac{1}{1\times4}+\dfrac{1}{4\times7}+...+\dfrac{1}{31\times34}\)
\(=\dfrac{1}{3}\cdot\left[3\cdot\left(\dfrac{1}{1\times4}+\dfrac{1}{4\times7}+...+\dfrac{1}{31\times34}\right)\right]\)
\(=\dfrac{1}{3}\cdot\left(\dfrac{3}{1\times4}+\dfrac{3}{4\times7}+...+\dfrac{3}{31\times34}\right)\)
\(=\dfrac{1}{3}\cdot\left(1-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{7}+...+\dfrac{1}{31}-\dfrac{1}{34}\right)\)
\(=\dfrac{1}{3}\cdot\left(1-\dfrac{1}{34}\right)\)
\(=\dfrac{1}{3}\cdot\dfrac{33}{34}\)
\(=\dfrac{11}{34}\)
1.b.1995/1996 và 11/12 = 1 - 1995/1996 và 1 - 11/12 = 1/1996 và 1/12
=1/1996 < 1/12 vậy 1995/1996 < 11/12
c.500/499 và 499/498 = 1 + 500/499 và 1 + 499/498 = 1/499 và 1/498
=1/499 < 1/498 vậy 500/499 < 499/498
2.a.1/9 x 10 + 1/10 x 11 + 1/11 x 12 + ... + 1/29 x 30
= 1/9 - 1/10 + 1/10 - 1/11 + 1/11 - 1/12 + ... + 1/29 - 1/30
=1/9 - 1/30 = 7/90
b.2/7 x 9 + 2/9 x 11 + 2/11 x 13 + ... + 2/37 x 39
=2 x (1/7 - 1/9 + 1/9 - 1/11 + 1/11 - 1/13 + ... + 1/37 - 1/39)
=2 x (1/7 - 1/39)
=64/275
c.1/1 x 4 + 1/4 x 7 + 1/7 x 10 + ... + 1/31 x 34
=1/1 - 1/4 + 1/4 - 1/7 + 1/7 - 1/10 + ... + 1/31 - 1/34
=1/1 - 1/34
=1/34