ĐẶT \(A=\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{2012}}\)
\(2A=1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{2011}}\)
\(2A-A=1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{2011}}-\left(\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{2012}}\right)\)
\(A=1-\dfrac{1}{2^{2012}}\)
1/ Chứng tỏ rằng : B=\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{8^2}< 1\)
2/ Rút gọn: B=\(\left(1-\dfrac{1}{2}\right)\cdot\left(1-\dfrac{1}{3}\right)\cdot\left(1-\dfrac{1}{4}\right)\cdot...\cdot\left(1-\dfrac{1}{20}\right)\)
3/ Tính giá trị của biểu thức: A= \(\dfrac{7}{4}\cdot\left(\dfrac{3333}{1212}+\dfrac{3333}{2020}+\dfrac{3333}{3030}+\dfrac{3333}{4242}\right)\)
4/ So sánh : A= \(\dfrac{2011+2012}{2010+2013}\) và B= \(\dfrac{2011}{2012}+\dfrac{2012}{2013}\)
Bài 1: Tính tổng
\(A=\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+...+\dfrac{1}{3^{100}}\)
cho tổng T= \(\dfrac{2}{2^1}+\dfrac{3}{2^2}+\dfrac{4}{2^3}\) +...+\(\dfrac{2016}{2^{2015}}+\dfrac{2017}{2^{2016}}\)
so sánh T với 3
Bài 2: Tính hợp lý:
\(A=\dfrac{63636337-37373763}{1+2+3+...+2006}\)
\(B=1\dfrac{6}{41}\left(\dfrac{12+\dfrac{12}{19}-\dfrac{12}{37}-\dfrac{12}{53}}{3+\dfrac{1}{3}-\dfrac{3}{37}-\dfrac{3}{53}}:\dfrac{4+\dfrac{4}{17}+\dfrac{4}{19}+\dfrac{4}{2006}}{5+\dfrac{5}{17}+\dfrac{5}{19}+\dfrac{5}{2006}}\right)\dfrac{124242423}{237373735}\)
Thực hiện phép tính sau:
\(\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{3^2}\right)\left(1-\dfrac{1}{4^2}\right)...\left(1-\dfrac{1}{10^2}\right)\).
a, \(\left(1+\dfrac{1}{2}\right)\left(1+\dfrac{1}{2^2}\right)\left(1+\dfrac{1}{2^3}\right)\left(1+\dfrac{1}{2^4}\right)....\left(1+\dfrac{1}{2^{50}}\right)< 3\)
b, \(\dfrac{1}{2}-\dfrac{1}{2^2}+.........+\dfrac{1}{2^{99}}-\dfrac{1}{2^{100}}< \dfrac{1}{3}\)
Câu 1. Cho A=\(\dfrac{1}{2^2}\)+\(\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{2012^2}\). So sánh A và 1.
Câu 2. Tính \(A=2014+\dfrac{2014}{1+2}+\dfrac{2014}{1+2+3}+\dfrac{2014}{1+2+3+4}+...+\dfrac{2014}{1+2+3+4+...+2013}\)
Câu 3. Cho A=\(\dfrac{6n+42}{6n}\)với n \(\in\) Z và n \(\ne\) 0. Tìm tất cả các số nguyên n sao cho A cũng là số nguyên.
Câu 4. So sánh A=\(\dfrac{17^{18}+1}{17^{19}+1}\) và B=\(\dfrac{17^{17}+1}{17^{18}+1}\).
Cho A = \(\dfrac{\left(3\dfrac{2}{15}+\dfrac{1}{5}\right):2\dfrac{1}{2}}{\left(5\dfrac{3}{7}-2\dfrac{1}{4}\right):4\dfrac{43}{56}}\) ; B = \(\dfrac{1,2:\left(1\dfrac{1}{5}.1\dfrac{1}{4}\right)}{0,32+\dfrac{2}{25}}\)
Chứng minh rằng A= B
Bài 2:Tìm x biết:
a)\(\dfrac{1}{7}+x=\dfrac{-2}{3}\)
b)\(\dfrac{-2}{3}:x=\dfrac{-5}{6}\)
c)\(\left\{\dfrac{3}{5}-2x\right\}.\dfrac{5}{8}=1\)
d)\(\dfrac{3}{4}+\dfrac{2}{5}x=\dfrac{29}{60}\)
e)\(\dfrac{3}{4}+\dfrac{1}{4}:x=\dfrac{2}{5}\)
f)\(\dfrac{11}{12}-\left(\dfrac{2}{5}+x\right)=\dfrac{2}{3}\)
g)\(\left|X+\dfrac{1}{3}\right|-4=\dfrac{-1}{2}\)
h)\(\left(\dfrac{1}{32}\right)^x.8^{2x}=512\)
i)\(5,3x+\left(-3,3\right)x+1,7=-4,9\)
