DN
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K
Khách
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MT
0
XO
5 tháng 2 2020
a) \(S=\left(-\frac{1}{7}\right)^0+\left(-\frac{1}{7}\right)^1+\left(-\frac{1}{7}\right)^2+...+\left(-\frac{1}{7}\right)^{2007}\)
\(=1+\left(-\frac{1}{7}\right)+\left(-\frac{1}{7}\right)^2+...+\left(-\frac{1}{7}\right)^{2007}\)
=> 7S = \(7+\left(-1\right)+\left(-\frac{1}{7}\right)+...+\left(-\frac{1}{7}\right)^{2006}\)
Lấy 7S trừ S ta có :
7S - S = \(7+\left(-1\right)+\left(-\frac{1}{7}\right)+...+\left(-\frac{1}{7}\right)^{2006}-\left[1+\left(-\frac{1}{7}\right)+\left(-\frac{1}{7}\right)^2+...+\left(-\frac{1}{7}\right)^{2007}\right]\)
6S = \(7-1-1+\left(\frac{1}{7}\right)^{2007}=5+\left(\frac{1}{7}\right)^{2007}\Rightarrow S=\frac{5+\left(\frac{1}{7}\right)^{2007}}{6}\)
HT
1
5 tháng 7 2015
gọi là A đi
\(A=1+\left(-\frac{1}{7}\right)^1+\left(-\frac{1}{7}\right)^2+...+\left(-\frac{1}{7}\right)^{2003}\Rightarrow\frac{-1}{7}A=-\frac{1}{7}+\left(-\frac{1}{7}\right)^2+...+\left(-\frac{1}{7}\right)^{2004}\)
=> \(-\frac{1}{7}A-A=-\frac{8}{7}A=\left[\left(-\frac{1}{7}\right)^1+\left(-\frac{1}{7}\right)^2+...+\left(-\frac{1}{7}\right)^{2004}\right]-\left[1+\left(-\frac{1}{7}\right)^1+\left(-\frac{1}{7}\right)^2+...+\left(-\frac{1}{7}\right)^{2003}\right]=-1+\left(-\frac{1}{7}\right)^{2004}\)
\(\Rightarrow A=\left(-1+\left(-\frac{1}{7}\right)^{2004}\right):-\frac{8}{7}\)
HT
1
MT
5 tháng 7 2015
(-1/7)0+(-1/7)1+(-1/7)2+...+(-1/7)2003
=1-1/7+1/7-1/7+....+1/7-1/7
=1
C
1
N
20 tháng 8 2017
\(\frac{1}{5^2}+\frac{1}{6^2}+......+\frac{1}{2007^2}>\frac{1}{5}\)
Có \(\frac{1}{5^2}>\frac{1}{4.5}\)
\(\frac{1}{6^2}>\frac{1}{5.6}\)
\(........\)
\(\frac{1}{2007^2}=\frac{1}{2006.2007}\)
\(\Rightarrow\frac{1}{5^2}+\frac{1}{6^2}+.......+\frac{1}{2007^2}< \frac{1}{4.5}+\frac{1}{5.6}+....+\frac{1}{2006.2007}\)
\(=\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+.....+\frac{1}{2006}-\frac{1}{2007}\)
\(=\frac{1}{4}-\frac{1}{2007}\)
\(=\frac{2003}{8028}>\frac{1}{5}\)
Đặt \(A=\left(\frac{-1}{7}\right)^0+\left(\frac{-1}{7}\right)^1+\left(\frac{-1}{7}\right)^2+...+\left(\frac{-1}{7}\right)^{2007}\)
\(\frac{-1}{7}.A=\left(\frac{-1}{7}\right)^1+\left(\frac{-1}{7}\right)^2+\left(\frac{-1}{7}\right)^3+...+\left(\frac{-1}{7}\right)^{2008}\)
\(A-\frac{-1}{7}.A=\left[\left(\frac{-1}{7}\right)^0+\left(\frac{-1}{7}\right)^1+\left(\frac{-1}{7}\right)^2+...+\left(\frac{-1}{7}\right)^{2007}\right]-\left[\left(\frac{-1}{7}\right)^1+\left(\frac{-1}{7}\right)^2+\left(\frac{-1}{7}\right)^3+...+\left(\frac{-1}{7}\right)^{2008}\right]\)
\(A+\frac{1}{7}.A=\left(\frac{-1}{7}\right)^0-\left(\frac{-1}{7}\right)^{2008}\)
\(\frac{8}{7}.A=1-\left(\frac{1}{7}\right)^{2008}\)
\(\frac{8}{7}.A=1-\frac{1}{7^{2008}}\)
\(A=\left(1-\frac{1}{7^{2008}}\right):\frac{8}{7}=\frac{\left(1-\frac{1}{7^{2008}}\right).7}{8}\)