CMR \(\frac{1}{7}+\frac{2}{7^2}+\frac{3}{7^3}+...+\frac{100}{7^{100}}< \frac{7}{36}\)
H24
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CMR : \(1+\frac{1}{3^1}+\frac{2}{3^2}+\frac{3}{3^3}+....+\frac{100}{3^{100}}< \frac{7}{4}\)\(\frac{7}{4}\)
Tính \(A=\left(36-\frac{36}{7^{100}}\right):\left(\frac{1}{7^1}+\frac{1}{7^2}+...+\frac{1}{7^{99}}+\frac{1}{7^{100}}\right)\)
Đặt \(E=\frac{1}{7}+\frac{1}{7^2}+\frac{1}{7^3}+...+\frac{1}{7^{99}}+\frac{1}{7^{100}}\)
\(\Rightarrow7E=1+\frac{1}{7}+\frac{1}{7^2}+...+\frac{1}{7^{98}}+\frac{1}{7^{99}}\)
\(\Rightarrow7E-E=\left(1+\frac{1}{7}+...+\frac{1}{7^{98}}+\frac{1}{7^{99}}\right)-\left(\frac{1}{7}+\frac{1}{7^2}+...+\frac{1}{7^{99}}+\frac{1}{7^{100}}\right)\)
\(\Rightarrow6E=1-\frac{1}{7^{100}}\)
\(\Rightarrow E=\frac{1-\frac{1}{7^{100}}}{6}\)
\(\Rightarrow A=\left(36-\frac{36}{7^{100}}\right):\frac{1-\frac{1}{7^{100}}}{6}\)
\(\Rightarrow A=36\left(1-\frac{1}{7^{100}}\right).\frac{6}{1-\frac{1}{7^{100}}}\)
\(\Rightarrow A=36.6=216\)
\(Tính.S=\left(-\frac{1}{7}\right)^0+\left(-\frac{1}{7}\right)^1+\left(-\frac{1}{7}\right)^2+...+\left(-\frac{1}{7}\right)^{2007}\)
\(CMR.\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+...+\frac{99}{100!}< 1\)
Cho \(Q=\frac{5}{7}.\frac{13}{7^2}.\frac{97}{7^4}....\frac{3^{2^{99}}+2^{2^{99}}}{7^{2^{99}}}\)CMR: \(Q.\left(7^{2^{100}-1}\right)\)
CMR:\(\frac{1}{7^2}-\frac{1}{7^4}+...+\frac{1}{7^{4n-2}}-\frac{1}{7^{4n}}+...+\frac{1}{98}-\frac{1}{100}< \frac{1}{50}\)
Cho M =\(\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}-\frac{4}{3^4}+...+\frac{99}{3^{99}}-\frac{100}{3^{100}}\) .Hãy chứng minh M<\(\frac{3}{16}\)
Câu 2 Chứng minh rằng :
\(\frac{1}{7^2}-\frac{1}{7^4}+...+\frac{1}{7^{98}}-\frac{1}{7^{100}}< \frac{1}{50}\)
Tham khảo nha bạn :
Câu hỏi của Trần Minh Hưng - Toán lớp | Học trực tuyến
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CMR :
\(\frac{1}{7^2}-\frac{1}{7^4}+\frac{1}{7^6}-\frac{1}{7^8}+...+\frac{1}{7^{98}}-\frac{1}{7^{100}}\)< \(\frac{1}{50}\)
Tính các tổng sau:
a) \(\frac{1}{7}+\frac{1}{7^2}+\frac{1}{7^3}+...+\frac{1}{7^{100}}.\)
b) \(-\frac{4}{5}+\frac{4}{5^2}-\frac{4}{5^3}+...+\frac{4}{5^{200}}.\)
c)\(\frac{-1}{3}+\frac{1}{3^2}-\frac{1}{3^3}+...+\frac{1}{3^{100}}-\frac{1}{3^{101}}\)
Đăt A = \(\frac{1}{7}+\frac{1}{7^2}+\frac{1}{7^3}+......+\frac{1}{7^{100}}\)
\(\Rightarrow7A=1+\frac{1}{7}+\frac{1}{7^2}+.....+\frac{1}{7^{100}}\)
\(\Rightarrow7A-A=1-\frac{1}{7^{100}}\)
\(\Rightarrow6A=1-\frac{1}{7^{100}}\)
\(\Rightarrow A=\frac{1-\frac{1}{7^{100}}}{6}\)
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Tính nhanh:
a) A=\(\frac{1}{3}-\frac{3}{4}-\left(-\frac{3}{5}\right)+\frac{1}{72}-\frac{2}{9}-\frac{1}{36}+\frac{2}{15}\)b) B=\(\frac{1}{5}-\frac{3}{7}+\frac{5}{9}-\frac{2}{11}+\frac{7}{13}-\frac{9}{16}-\frac{7}{13}+\frac{2}{11}-\frac{5}{9}+\frac{3}{7}-\frac{1}{5}\)c) C= \(\frac{1}{100}-\frac{1}{100.99}-\frac{1}{99.98}-...-\frac{1}{3.2}-\frac{1}{2.1}\)
