Ta có: \(2P=2+\dfrac{2}{2^0}+\dfrac{3}{2^1}+...+\dfrac{1992}{2^{1990}}\)
\(\Rightarrow2P-P=\left(2+\dfrac{2}{2^0}+\dfrac{3}{2^1}+...+\dfrac{1992}{2^{1990}}\right)-\left(\dfrac{1}{2^0}+\dfrac{2}{2^1}+...+\dfrac{1992}{2^{1991}}\right)\)
\(=2-\dfrac{1992}{2^{1991}}+\left(\dfrac{1}{2^0}+\dfrac{1}{2^1}+...+\dfrac{1}{2^{1990}}\right)\)
\(=2-\dfrac{1992}{2^{1991}}+2-\dfrac{1}{2^{1990}}< 4\)
Cho \(A=1+\dfrac{3}{2^3}+\dfrac{4}{2^4}+\dfrac{5}{2^5}+...+\dfrac{100}{2^{100}}\). Chứng minh A < 2.
B1Tìm x,y biết:
\(\dfrac{x}{2}=\dfrac{y}{7}\) và 2x-5y=93
B2Cho 4 số a1, a2, a3. a4 khác 0 và thỏa mãn :
\(a\dfrac{2}{2}=a1\cdot a3\) và \(a\dfrac{2}{3}=a2\cdot a4\)
Chứng minh rằng:\(\dfrac{a\dfrac{3}{1}+a\dfrac{3}{2}+a\dfrac{3}{3}}{a\dfrac{3}{2}+a\dfrac{3}{3}+a\dfrac{3}{4}}=\dfrac{a1}{a4}\)
Chứng minh rằng : \(B=\dfrac{1}{3}+\dfrac{2}{3^2}+\dfrac{3}{3^3}+\dfrac{4}{3^4}+...+\dfrac{100}{3^{100}}< \dfrac{3}{4}\)
tìm x biết
a) \(\dfrac{1}{2}x+2\dfrac{1}{2}=3\dfrac{1}{2}x-\dfrac{3}{4}\)
b) \(\dfrac{2}{3}x-\dfrac{2}{5}=\dfrac{1}{2}x-\dfrac{1}{3}\)
c) \(\dfrac{1}{3}x+\dfrac{2}{5}\left(x+1\right)=0\)
d) \(\dfrac{2}{3}-\dfrac{1}{3}\left(x-\dfrac{3}{2}\right)-\dfrac{1}{2}\left(2x+1\right)=5\)
e) \(\dfrac{x+2015}{5}+\dfrac{x+2016}{4}=\dfrac{x+2017}{3}+\dfrac{x+2018}{2}\)
Chứng minh rằng:
\(\dfrac{1}{6}< \dfrac{1}{5^2}+\dfrac{1}{6^2}+\dfrac{1}{7^2}+...+\dfrac{1}{100^2}< \dfrac{1}{4}\)
Chứng minh rằng
\(\dfrac{1}{6}< \dfrac{1}{5^2}+\dfrac{1}{6^2}+\dfrac{1}{7^2}+...+\dfrac{1}{100^2}< \dfrac{1}{4}\)
1) Tìm a, b > 0
Sao cho \(\dfrac{1}{a}-\dfrac{1}{b}=\dfrac{1}{a-b}\)
2) Tính nhanh:
\(-4\dfrac{1}{3}\left(\dfrac{1}{2}-\dfrac{1}{6}\right)\le x\le\dfrac{-2}{3}\left(\dfrac{1}{3}-\dfrac{1}{2}-\dfrac{3}{4}\right)\)
Chứng minh rằng : \(A=\dfrac{3}{1^2.2^2}+\dfrac{5}{2^2.3^2}+\dfrac{7}{3^2.4^2}+...+\dfrac{19}{9^2.10^2}< 1\)
Chứng minh rằng:
\(\dfrac{1}{7^2}-\dfrac{1}{7^4}+...+\dfrac{1}{7^{4n-2}}-\dfrac{1}{7^{4n}}+...+\dfrac{1}{7^{98}}+\dfrac{1}{7^{100}}< \dfrac{1}{50}\)
