CMR:
a) \(\dfrac{1}{2^2}+\dfrac{1}{4^2}+\dfrac{1}{6^2}+...+\dfrac{1}{100^2}< \dfrac{1}{2}\)
b) \(\dfrac{3}{4}+\dfrac{8}{9}+\dfrac{15}{16}+...+\dfrac{2499}{2500}>48\)
BT1: CMR:
a) \(\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{n^2}< 1\)
b) \(\dfrac{1}{4}+\dfrac{1}{16}+\dfrac{1}{36}+\dfrac{1}{64}+\dfrac{1}{100}+\dfrac{1}{144}+\dfrac{1}{196}< \dfrac{1}{2}\)
c) \(\dfrac{1}{3}+\dfrac{1}{30}+\dfrac{1}{32}+\dfrac{1}{35}+\dfrac{1}{45}+\dfrac{1}{47}+\dfrac{1}{50}< \dfrac{1}{2}\)
d) \(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{8}-\dfrac{1}{16}+\dfrac{1}{32}-\dfrac{1}{64}< \dfrac{1}{3}\)
e) \(\dfrac{1}{3}< \dfrac{2}{3^2}+\dfrac{3}{3^3}-\dfrac{4}{3^4}+...+\dfrac{99}{3^{99}}-\dfrac{100}{3^{100}}< \dfrac{3}{16}\)
f) \(\dfrac{1}{41}+\dfrac{1}{42}+\dfrac{1}{43}+...+\dfrac{1}{79}+\dfrac{1}{80}>\dfrac{7}{12}\)
BT2: Tính tổng
a) A=\(\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+...+\dfrac{1}{3^{100}}\)
b) E=\(1+\dfrac{1}{2}\left(1+2\right)+\dfrac{1}{3}\left(1+2+3\right)+\dfrac{1}{4}\left(1+2+3+4\right)+...+\dfrac{1}{200}\left(1+2+3+...+200\right)\)
BT3: Cho S=\(\dfrac{3}{10}+\dfrac{3}{11}+\dfrac{3}{12}+\dfrac{3}{13}+\dfrac{3}{14}\)
CMR: 1 < S < 2
tính nhanh
2155-(174+2155)+(-68+174)
2.\(\dfrac{3}{7}\left(\dfrac{2}{9}-1\dfrac{3}{7}\right)-\dfrac{5}{3}:\dfrac{1}{9}\)
\(\left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)\left(1-\dfrac{1}{4}\right)\left(1-\dfrac{1}{5}\right)\)
\(\left(\dfrac{377}{-231}-\dfrac{123}{89}+\dfrac{34}{791}\right).\left(\dfrac{1}{6}-\dfrac{1}{8}-\dfrac{1}{24}\right)\)
chứng tỏ phân số sau tối giản vs mọi số tự nhiên n\(\dfrac{n+1}{2n+3}\)
1. Tính giá trị của biểu thức :
a) ( 1 - \(\dfrac{1}{3}\) ) . ( 1- \(\dfrac{1}{6}\) ) . ( 1 - \(\dfrac{1}{10}\) ) ...... ( 1 - \(\dfrac{1}{780}\) )
b) \(\dfrac{2^4}{7.15}+\dfrac{2^4}{15.23}+\dfrac{2^4}{23.31}+.....+\dfrac{2^4}{55.63}-\dfrac{6.\left(-14\right)-17.\left(-7\right).\left(-2\right)}{-22.28}\)
2. Tìm số tự nhiên n biết : \(\dfrac{4}{3.5}+\dfrac{8}{5.9}+\dfrac{12}{9.15}+.....+\dfrac{32}{n.\left(n+16\right)}=\dfrac{16}{25}\)
3. So sánh A và B biết :
a) A = \(\dfrac{2003.2004-1}{2003.2004}\) và B = \(\dfrac{2004.2005-1}{2004.2005}\)
b) A = \(\dfrac{10^{25}+1}{10^{26}+1}\) và B = \(\dfrac{10^{26}+1}{10^{27}+1}\)
bài 7 Tìm phân số tiếp theo
\(3;4\dfrac{1}{2};6\dfrac{3}{4};10\dfrac{1}{8};15\dfrac{3}{16};?\)
cho S=\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+....+\dfrac{1}{9^2}\)
chứng minh rằng \(\dfrac{2}{5}\)<S<\(\dfrac{8}{9}\)
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) \(C=\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{16}\)
Câu 9.Thực hiện phép tính:
M=\(\dfrac{-3}{4}.\dfrac{2}{11}+\dfrac{-3}{4}.\dfrac{9}{11}+2\dfrac{3}{4}\)
N=\(\dfrac{6}{8}+\dfrac{5}{8}:5-\dfrac{3}{16}\).(\(-4^2\))
Cho S = \(\dfrac{1}{4^2}+\dfrac{1}{5^2}+\dfrac{1}{6^2}+.......+\dfrac{1}{50^2}\) Chứng minh S không là số nguyên
