So sanh hai so :(1/2)^30va(1/3)^20
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Ta có: \(S=1+3+3^2+...+3^{20}\)
\(\Rightarrow3S=3+3^2+3^3+...+3^{21}\)
\(\Rightarrow3S-S=\left(3+3^2+3^3+...+3^{21}\right)-\left(1+3+3^2+...+3^{20}\right)\)
\(\Rightarrow2S=3^{21}-1\)
\(\Rightarrow S=\left(3^{21}-1\right).\frac{1}{2}\)
\(\Rightarrow S=3^{21}.\frac{1}{2}-\frac{1}{2}\)
Vì \(3^{21}.\frac{1}{2}-\frac{1}{2}< 3^{21}.\frac{1}{2}\) nên \(A< \frac{1}{2}.3^{21}\)
Vậy \(A< \frac{1}{2}.3^{21}\)
Ta thấy B=20^10-1/20^10-3 là phân số lớn hơn 1.
Theo tính chất nếu a/b>1 thì a/b > a+n/b+n ( n khác 0 )
Ta có : 20^10-1/20^10-3 > 20^10-1+2/20^10-3+2
<=> B > 20^10+1/20^10-3 = A
<=> B > A
Vậy B > A
Ta có công thức sau: \(\frac{a}{b}\) > \(\frac{a+m}{b+m}\) ( m khác 0;\(\frac{a}{b}\)>1)
Vì 20^10-1>20^10-3 => B>1
Áp dụng vào bài giải ta có:
A=\(\frac{\left(20^{10}-1\right)+2}{\left(20^{10}-3\right)+2}\) < \(\frac{20^{10}-1}{20^{10}-3}\)= B
Vậy A < B
\(A=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)...\left(1-\frac{1}{19}\right)\left(1-\frac{1}{20}\right)\)
\(A=\left(\frac{2}{2}-\frac{1}{2}\right)\left(\frac{3}{3}-\frac{1}{3}\right)...\left(\frac{19}{19}-\frac{1}{19}\right)\left(\frac{20}{20}-\frac{1}{20}\right)\)
\(A=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}...\frac{18}{19}.\frac{19}{20}\)
\(A=\frac{1.2.3...18.19}{2.3.4...19.20}\)
\(A=\frac{1}{20}\Leftrightarrow A>\frac{1}{21}\)
\(A=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right).....\left(1-\frac{1}{20}\right)\)
\(A=\frac{1}{2}.\frac{2}{3}......\frac{19}{20}=\frac{1}{20}>\frac{1}{21}\)
\(\text{Vậy: A lớn hơn 1/21}\)
\(\Rightarrow\)2K=\(1+\frac{1}{2}+\frac{1}{2^2}+....+\frac{1}{2^{19}}\)\(\Rightarrow2K-k=k=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{19}}-k\)
\(\Rightarrow k=1-\frac{1}{2^{20}}< 1\)
\(\Rightarrow k< H\)
Vậy......
\(\left(\frac{1}{2}\right)^{30}=\left(\frac{1}{2}^3\right)^{10}\)
\(\Rightarrow\left(\frac{1}{2}^3\right)^{10}=\frac{1}{6}^{10}\)
\(\frac{1}{3}^{20}=\left(\frac{1}{3}^2\right)^{10}=\frac{1}{9}^{10}\)
\(\frac{1}{6}^{10}>\frac{1}{9}^{10}\Rightarrow\left(\frac{1}{2}\right)^{30}>\left(\frac{1}{3}\right)^{20}\)
(1/2)^30=(1/2^3)^10=(1/8)^10
(1/3)^20=(1/2^2)^10=(1/4)^10
Vì 1/8 <1/4 nên (1/8)^10<(1/4)^10
=>(1,2)^30<(1/3)^20
click cho mk nha