Cho\(A=\frac{1}{5}+\frac{2}{5^2}+\frac{3}{5^3}+...+\frac{11}{5^{11}}\).Chứng minh:\(A< \frac{5}{16}\)
Cho \(A=\frac{1}{5^2}+\frac{2}{5^3}+\frac{3}{5^4}+...+\frac{n}{5^{n+1}}+...+\frac{11}{5^{12}}\)
Chứng minh A<\(\frac{1}{16}\)
CHỨNG MINH RẰNG:
A = \(\frac{1}{5^2}+\frac{2}{5^3}+\frac{3}{5^4}+...+\frac{11}{5^{12}}< \frac{1}{16}\)
5A=1/5=2/5^2+......+11/5^11
4A=1/5+1/5^2+......+1/5^11-11/5^12
20A=1+1/5+1/5^2+.....+1/5^10-11/5^11
16A=1-1/5^11+11/5^12-11/5^11
vi 1-1/5^11<1;11/5^12-11/5^11<0
16A<1
A<1/16
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Cho A=\(\frac{1}{5}\)+\(\frac{2}{5^2}\)+\(\frac{3}{5^3}\)+...+\(\frac{10}{5^{10}}\)+\(\frac{11}{5^{11}}\).Chứng minh A<\(\frac{5}{16}\)
giải bài này hộ mk nha
Cho A= \(\frac{1}{5}\)+\(\frac{2}{5^2}\)+\(\frac{3}{5^3}\)+....+\(\frac{10}{5^{10}}\)+\(\frac{11}{5^{11}}\).Chứng minh A<\(\frac{5}{16}\)
\(A=\frac{1}{5}+\frac{2}{5^2}+\frac{3}{5^3}+....+\frac{10}{5^{10}}+\frac{11}{5^{11}}\)
\(\Rightarrow5A=1+\frac{2}{5}+\frac{3}{5^2}+....+\frac{10}{5^9}+\frac{11}{5^{10}}\)
\(\Rightarrow5A-A=\left(1+\frac{2}{5}+...+\frac{11}{5^{10}}\right)-\left(\frac{1}{5}+\frac{2}{5^2}+...+\frac{11}{5^{11}}\right)\)
\(\Rightarrow4A=1+\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{10}}-\frac{11}{5^{11}}\)(1)
Đặt \(B=1+\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{10}}\)
\(\Rightarrow5B=5+1+\frac{1}{5}+...+\frac{1}{5^9}\)
\(\Rightarrow5B-B=\left(5+1+...+\frac{1}{5^9}\right)-\left(1+\frac{1}{5}+...+\frac{1}{5^{10}}\right)\)
\(\Rightarrow4B=5-\frac{1}{5^{10}}< 5\)
\(\Rightarrow B< \frac{5}{4}\)(2)
Thay (2) vào (1) \(\Rightarrow4A< \frac{5}{4}-\frac{11}{5^{11}}< \frac{5}{4}\)
\(\Rightarrow A< \frac{5}{16}\left(đpcm\right)\)
Cho A= \(\frac{1}{5^2}+\frac{2}{5^3}+....+\frac{n}{5^{n+1}}+.....+\frac{11}{5^{12}}\)
Chứng minh rằng A < \(\frac{1}{16}\)
Cho A=\(\frac{1}{5^2}+\frac{2}{5^3}+\frac{3}{5^4}+...+\frac{n}{5^{n+1}}+...+\frac{11}{5^{12}}\)với n\(\inℕ\).Chứng minh rằng A<\(\frac{1}{16}\)
Giúp mình với, hiện đang cần gấp lắm.
5A=\(\frac{1}{5}+\frac{2}{5^2}...+\frac{n}{5^n}...+\frac{11}{5^{11}}\)
=>4A=5A-A=\(\frac{1}{5}+\frac{1}{5^2}...+\frac{1}{5^{11}}-\frac{11}{5^{12}}\)
=>20A=\(1+\frac{1}{5}+...+\frac{1}{5^{10}}-\frac{11}{5^{11}}\)
=>16A=20A-4A=\(1-\frac{1}{5^{11}}+\frac{11}{5^{12}}-\frac{11}{5^{11}}\)
Mà \(1-\frac{1}{5^{11}}< 1\),\(\frac{11}{5^{12}}-\frac{11}{5^{11}}< 0\)
=>16A<1
Do đó: A<1/16(đpcm)
Chứng minh: \(\frac{1}{5^2}+\frac{2}{5^3}+\frac{3}{5^4}+...+\frac{11}{5^{12}}<\frac{1}{16}\)
\(\left(\frac{1}{5^2}+\frac{2}{5^3}+.....+\frac{11}{5^{12}}\right)\)
=\(\left(\frac{1}{5^2}+\frac{2}{5^3}+.....+\frac{11}{5^{12}}\right)\)<\(\frac{1}{4.5}+\frac{2}{4.5.6}+...+\frac{11}{4.5.6...15}\)
=???
Chứng minh:
\(\frac{1}{5^2}+\frac{2}{5^3}+\frac{3}{5^4}+...+\frac{11}{5^{12}}<\frac{1}{16}\)
Cho A = \(\frac{1}{5^2}+\frac{1}{5^3}+\frac{1}{5^4}+.......+\frac{n}{5^{n+1}}+.......+\frac{11}{5^{12}}\) với n \(\in\) N. Chứng minh rằng A < \(\frac{1}{16}\)
Giúp mk vs