Bài 2 : Cho S = \(\dfrac{3}{10}+\dfrac{3}{11}+\dfrac{3}{12}+\dfrac{3}{14}+\dfrac{3}{15}\)
Chứng tỏ 1<S<2
Cho S= \(\dfrac{3}{10}+\dfrac{3}{11}+\dfrac{3}{12}+\dfrac{3}{13}+\dfrac{3}{14}\)
Chứng minh rằng: 1<S<2
Ta có: \(\dfrac{3}{10}>\dfrac{3}{15}\)
\(\dfrac{3}{11}>\dfrac{3}{15}\)
\(\dfrac{3}{12}>\dfrac{3}{15}\)
\(\dfrac{3}{13}>\dfrac{3}{15}\)
\(\dfrac{3}{14}>\dfrac{3}{15}\)
Do đó: \(\dfrac{3}{10}+\dfrac{3}{11}+\dfrac{3}{12}+\dfrac{3}{13}+\dfrac{3}{14}>\dfrac{3}{15}+\dfrac{3}{15}+\dfrac{3}{15}+\dfrac{3}{15}+\dfrac{3}{15}=1\)
hay 1<S(1)
Ta có: \(\dfrac{3}{11}< \dfrac{3}{10}\)
\(\dfrac{3}{12}< \dfrac{3}{10}\)
\(\dfrac{3}{13}< \dfrac{3}{10}\)
\(\dfrac{3}{14}< \dfrac{3}{10}\)
Do đó: \(\dfrac{3}{11}+\dfrac{3}{12}+\dfrac{3}{13}+\dfrac{3}{14}< \dfrac{3}{10}+\dfrac{3}{10}+\dfrac{3}{10}+\dfrac{3}{10}=\dfrac{12}{10}\)
\(\Leftrightarrow S< \dfrac{15}{10}=\dfrac{3}{2}< 2\)(2)
Từ (1) và (2) suy ra 1<S<2(đpcm)
>, <, = ?
\(\dfrac{4}{11}\) ... \(\dfrac{6}{11}\) \(\dfrac{6}{7}\) ... \(\dfrac{12}{14}\)
\(\dfrac{15}{17}\) ... \(\dfrac{10}{17}\) \(\dfrac{2}{3}\) ... \(\dfrac{3}{4}\)
\(\dfrac{4}{11}< \dfrac{6}{11}\)
\(\dfrac{6}{7}=\dfrac{12}{14}\)
\(\dfrac{15}{17}>\dfrac{10}{17}\)
\(\dfrac{2}{3}< \dfrac{3}{4}\)
4/11 < 6/11
15/17 > 10/17
6/7 = 12/14
2/3< 3/4
1) Cho S = \(\dfrac{3}{10}+\dfrac{3}{11}+\dfrac{3}{12}+\dfrac{3}{13}+\dfrac{3}{14}\)
Chứng minh rằng 1 < S < 2 từ đó suy ra S không phải là số tự nhiên
\(S=\dfrac{3}{10}+\dfrac{3}{11}+\dfrac{3}{12}+\dfrac{3}{13}+\dfrac{3}{14}\)
\(S=\left(\dfrac{3}{10}+\dfrac{3}{12}+\dfrac{3}{14}\right)+\left(\dfrac{3}{11}+\dfrac{3}{13}\right)\)
Đến bước trên thì do mình lười đánh máy nên bạn tính trong ngoặc bằng máy tính thì sẽ ra kết quả dưới đây (làm tắt):
\(S=\dfrac{107}{140}+\dfrac{72}{143}\)
Bước này phải quy đồng nhé! Ra số hơi dài nhưng phải chịu thôi bạn!
\(S=1,267782218\)
Mà \(1< 1,267782218< 2\)
Suy ra \(1< S< 2\)
Suy ra Điều phải chứng minh.
Xong rồi bạn, tick ''Đúng'' cho mình nhé!
\(\text{Bài 4. Chứng tỏ rằng:}\)
\(a\)) \(\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{30^2}< 1\)
\(b\)) \(\dfrac{1}{10}+\dfrac{1}{11}+\dfrac{1}{12}+...+\dfrac{1}{99}+\dfrac{1}{100}>1\)
\(c\)) \(\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+...+\dfrac{1}{17}< 2\)
\(d\)) \(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{29.30}< 1\)
a)
\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{30^2}\\ < \dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{29.30}\\ =1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{29}-\dfrac{1}{30}\\ =1-\dfrac{1}{30}=\dfrac{29}{30}< 1\left(dpcm\right)\)
b)
\(\dfrac{1}{10}+\dfrac{1}{11}+\dfrac{1}{12}+...+\dfrac{1}{99}+\dfrac{1}{100}=\dfrac{1}{10}+\left(\dfrac{1}{11}+\dfrac{1}{12}+...+\dfrac{1}{99}+\dfrac{1}{100}\right)\\ >\dfrac{1}{10}+\dfrac{1}{100}+\dfrac{1}{100}+...+\dfrac{1}{100}=\dfrac{1}{10}+\dfrac{90}{100}\\ =\dfrac{110}{100}>1\left(đpcm\right).\)
c)
\(\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+...+\dfrac{1}{17}\\ =\left(\dfrac{1}{5}+\dfrac{1}{6}+...+\dfrac{1}{9}\right)+\left(\dfrac{1}{10}+\dfrac{1}{11}+...+\dfrac{1}{17}\right)\\ < \dfrac{1}{5}.5+\dfrac{1}{8}.8=1+1=2\left(đpcm\right)\)
d) tương tự câu 1
\(\dfrac{2}{5}\)*\(15\dfrac{1}{3}\)-\(\dfrac{2}{5}\)*\(10\dfrac{1}{3}\)
\(12\dfrac{5}{11}\)-(\(3\dfrac{1}{4}\)+\(2\dfrac{5}{11}\)]
\(\dfrac{34}{31}\)-\(\dfrac{19}{28}\)-\(\dfrac{3}{31}\)
\(\dfrac{2}{5}\times15\dfrac{1}{3}-\dfrac{2}{5}\times10\dfrac{1}{3}\)
\(=\dfrac{2}{5}\times\left(15\dfrac{1}{3}-10\dfrac{1}{3}\right)\)
\(=\dfrac{2}{5}\times5\)
\(=2\)
____________________
\(12\dfrac{5}{11}-\left(3\dfrac{1}{4}+2\dfrac{5}{11}\right)\)
\(=12\dfrac{5}{11}-3\dfrac{1}{4}-2\dfrac{5}{11}\)
\(=\left(12\dfrac{5}{11}-2\dfrac{5}{11}\right)-3\dfrac{1}{4}\)
\(=10-3\dfrac{1}{4}\)
\(=\dfrac{27}{4}\)
_______________
\(\dfrac{34}{31}-\dfrac{19}{28}-\dfrac{3}{31}\)
\(=\left(\dfrac{34}{31}-\dfrac{3}{31}\right)-\dfrac{19}{28}\)
\(=\dfrac{31}{31}-\dfrac{19}{28}\)
\(=1-\dfrac{19}{28}\)
\(=\dfrac{9}{28}\)
1. Tính giá trị biểu thức:
M=\(1+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{100}}\)
2.Chứng tỏ rằng :
\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{100^2}< 1\)
3.So sánh
a, A=\(\dfrac{11^5+1}{11^6+1}\) ; B=\(\dfrac{11^6+1}{11^7+1}\)
b, M=\(\dfrac{15^{10}-1}{15^9+1}\) ; N=\(\dfrac{15^9-1}{15^{10}+1}\)
mọi người thật là nhẫn tâm
chẳng ai giúp mk
TRỜI ƠI!!! AI MS LÀ BN BÈ THỰC SỰ
Ko cs đứa mô trả lời chứ chi
Loại bn bè vs mấy ng chỉ là giả tạo thôi
Bài 1:Tính hợp lí
a)-12,5+17,55-3,5+2,45
b)0,175-(\(2\dfrac{1}{3}\)+0,175)
c)\(\dfrac{5}{13}\).\(\dfrac{-3}{10}\)+\(\dfrac{3}{10}\).\(\dfrac{-8}{13}\)+(-0,7)
Bài 2:Tìm x biết
a)x+\(\dfrac{2}{5}\)=2,4
b)2x-\(\dfrac{4}{5}\)=-1,5
c)11-(15+11)=x-(25-9)
Bài 3:Cho A=\(\dfrac{1}{2^2}\)+\(\dfrac{1}{3^2}\)+\(\dfrac{1}{4^2}\)+\(\dfrac{1}{5^2}\)+....+\(\dfrac{1}{100^2}\)
Chứng tỏ A<1
2:
a: x=2,4-0,4=2
b: =>2x=-1,5+0,8=-0,7
=>x=-0,35
c: =>x-16=-15
=>x=1
tính
a)\(\dfrac{-10}{11}.\dfrac{8}{9}+\dfrac{7}{18}.\dfrac{10}{11}\)
b)\(\dfrac{3}{14}:\dfrac{1}{28}-\dfrac{13}{21}:\dfrac{1}{28}+\dfrac{29}{42}:\dfrac{1}{28}-8\)
c)\(-1\dfrac{5}{7}.15+\dfrac{2}{7}\left(-15\right)+\left(-105\right).\left(\dfrac{2}{3}-\dfrac{4}{5}+\dfrac{1}{7}\right)\)
ăn lồn cái địt mẹ mày lửa trùa ĐẦU LỒN nhá
Bài 1.( 2 điểm)Tính bằng cách hợp lí:
a) \(\dfrac{1}{2}+\dfrac{9}{10}+\dfrac{5}{6}-\dfrac{11}{14}-\dfrac{1}{3}+\dfrac{-4}{35}\)
b) \(\left(\dfrac{18}{23}+\dfrac{7}{12}\right)+\left(\dfrac{-13}{19}-\dfrac{3}{4}\right)+\left(\dfrac{-6}{19}+\dfrac{5}{23}\right)\)
c) \(\dfrac{4}{3}+\dfrac{-5}{6}+\dfrac{-1}{4}\)
d) \(\dfrac{5}{6}-\dfrac{7}{5}+\dfrac{17}{30}\)
1: \(\dfrac{1}{2}+\dfrac{9}{10}+\dfrac{5}{6}-\dfrac{11}{14}-\dfrac{1}{3}+\dfrac{-4}{35}\)
\(=\left(\dfrac{1}{2}+\dfrac{5}{6}-\dfrac{1}{3}\right)+\dfrac{9}{10}-\left(\dfrac{11}{14}+\dfrac{4}{35}\right)\)
\(=\dfrac{3+5-2}{6}+\dfrac{9}{10}-\dfrac{55+8}{70}\)
\(=1+\dfrac{9}{10}-\dfrac{9}{10}\)
=1