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9 tháng 7 2021

Đặt S = \(\frac{1}{2}+\frac{1}{2^5}+\frac{1}{2^9}+...+\frac{1}{2^{101}}\)

=> 24S = 16S = \(2^3+\frac{1}{2}+\frac{1}{2^5}+...+\frac{1}{2^{97}}\)

=> 16S - S = \(2^3+\frac{1}{2}+\frac{1}{2^5}+...+\frac{1}{2^{97}}-\left(\frac{1}{2}+\frac{1}{2^5}+\frac{1}{2^9}+...+\frac{1}{2^{101}}\right)\)

=> 15S = \(2^3-\frac{1}{2^{101}}\)

=> S = \(\frac{2^3-\frac{1}{2^{101}}}{15}\)

Khi đó A = \(\frac{2^3-\frac{1}{2^{101}}}{15}:\left(2^3-\frac{1}{2^{101}}\right)=\frac{1}{15}\)

9 tháng 7 2021

kết bạn đi toán lớp mấy vậy

Đặt \(A=\frac{1}{5}+\frac{1}{5^3}+...+\frac{1}{5^{101}}\)

\(\Rightarrow25A=5+\frac{1}{5}+\frac{1}{5^3}+...+\frac{1}{5^{99}}\)

\(\Rightarrow25A-A=\left(5+\frac{1}{5}+\frac{1}{5^3}+...+\frac{1}{5^{99}}\right)-\left(\frac{1}{5}+\frac{1}{5^3}+\frac{1}{5^5}+...+\frac{1}{5^{101}}\right)\)

hay \(24A=5-\frac{1}{5^{101}}\)

\(\Rightarrow A=\frac{5-\frac{1}{5^{101}}}{24}\)

\(\Rightarrow A:\left(1-\frac{1}{5^{102}}\right)=\frac{5-\frac{1}{5^{101}}}{24}.\frac{1}{1-\frac{1}{5^{102}}}\)

\(=\frac{5\left(1-\frac{1}{5^{102}}\right)}{24}.\frac{1}{1-\frac{1}{5^{102}}}=\frac{5}{24}\)

9 tháng 7 2021

Đặt S = \(\frac{1}{7^2}+\frac{1}{7^4}+\frac{1}{7^6}+...+\frac{1}{7^{100}}\)

=> 72S = 49S = \(1+\frac{1}{7^2}+\frac{1}{7^4}+...+\frac{1}{7^{98}}\)

=> 49S - S = \(\left(1+\frac{1}{7^2}+\frac{1}{7^4}+...+\frac{1}{7^{98}}\right)-\left(\frac{1}{7^2}+\frac{1}{7^4}+\frac{1}{7^6}+...+\frac{1}{7^{100}}\right)\)

=> 48S = \(1-\frac{1}{7^{100}}\)

=> \(S=\frac{1-\frac{1}{7^{100}}}{48}\)

Khi đó A = \(\left(\frac{1-\frac{1}{7^{100}}}{48}\right):\left(1-\frac{1}{7^{100}}\right)=\frac{1}{48}\)

\(-4\frac{1}{3}.\left(\frac{1}{2}-\frac{1}{6}\right)\le x\le-\frac{2}{3}.\left(\frac{1}{3}-\frac{1}{2}-\frac{3}{4}\right)\)

\(\Rightarrow-\frac{13}{3}.\left(\frac{3}{6}-\frac{1}{6}\right)\le x\le-\frac{2}{3}.\left(\frac{4}{12}-\frac{6}{12}-\frac{9}{12}\right)\)

\(\Rightarrow-\frac{13}{3}.\frac{2}{6}\le x\le-\frac{2}{3}.\frac{-11}{12}\)

\(\Rightarrow\frac{-13}{9}\le x\le\frac{11}{18}\)

\(\Rightarrow\frac{-26}{18}\le x\le\frac{11}{18}\)

=> -1,44444444444........... ≤ x ≤ 0,6111111111...........

Mà x ∈ Z

=> x ∈ { -1 ; 0 }

14 tháng 7 2021

\(x\in\varnothing\) 

14 tháng 7 2021

khong biet

8 tháng 7 2021

Sửa đề \(\frac{3}{2}+\frac{5}{2^2}+\frac{9}{2^3}+...+\frac{2^{100}+1}{2^{100}}=\frac{2+1}{2}+\frac{2^2+1}{2^2}+\frac{2^3+1}{2^3}+...+\frac{2^{100}+1}{2^{100}}\)

\(\left(1+1+1+...+1\right)+\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{100}}\right)\)(100 hạng tử 1) 

\(100+\left(1-\frac{1}{2^{100}}\right)=101-\frac{1}{2^{100}}< 101\)(1)

Vì \(-\frac{1}{2^{100}}>-1\Rightarrow101-\frac{1}{2^{100}}>101-1\Rightarrow B>100\)(2)

Từ (1) và (2) => 100 < B < 101 

DD
9 tháng 7 2021

\(A=\frac{1}{5}+\frac{1}{5^3}+\frac{1}{5^5}+...+\frac{1}{5^{101}}\)

\(\frac{1}{5^2}A=\frac{1}{5^3}+\frac{1}{5^5}+\frac{1}{5^7}+...+\frac{1}{5^{103}}\)

\(\left(1-\frac{1}{5^2}\right)A=\left(\frac{1}{5}+\frac{1}{5^3}+\frac{1}{5^5}+...+\frac{1}{5^{101}}\right)-\left(\frac{1}{5^3}+\frac{1}{5^5}+\frac{1}{5^7}+...+\frac{1}{5^{103}}\right)\)

\(\frac{24}{25}A=\frac{1}{5}-\frac{1}{5^{103}}\)

\(A=\left(1-\frac{1}{5^{102}}\right).\frac{5}{24}\)

Suy ra \(\left(\frac{1}{5}+\frac{1}{5^3}+\frac{1}{5^5}+...+\frac{1}{5^{101}}\right)\div\left(1-\frac{1}{5^{102}}\right)=\frac{5}{24}\).

DD
14 tháng 7 2021

\(\frac{2}{1.4.7}+\frac{2}{4.7.10}+...+\frac{2}{58.61.64}\)

\(=\frac{1}{3}\left(\frac{6}{1.4.7}+\frac{6}{4.7.10}+...+\frac{6}{58.61.64}\right)\)

\(=\frac{1}{3}\left(\frac{7-1}{1.4.7}+\frac{10-4}{4.7.10}+...+\frac{64-58}{58.61.64}\right)\)

\(=\frac{1}{3}\left(\frac{1}{1.4}-\frac{1}{4.7}+\frac{1}{4.7}-\frac{1}{7.10}+...+\frac{1}{58.61}-\frac{1}{61.64}\right)\)

\(=\frac{1}{3}\left(\frac{1}{1.4}-\frac{1}{61.64}\right)\)

\(=\frac{325}{3904}\)

\(\frac{2}{1.4.7}+\frac{2}{4.7.10}+...+\frac{2}{58.61.64}\)

\(=\frac{1}{3}.\left(\frac{6}{1.4.7}+\frac{6}{4.7.10}+...+\frac{6}{58.61.64}\right)\)

\(=\frac{1}{3}.\left(\frac{7 - 1}{1.4.7}+\frac{10 - 4}{4.7.10}+...+\frac{64 - 58}{58.61.64}\right)\)

\(=\frac{1}{3}.\left(\frac{1}{4}-\frac{1}{4.7}+\frac{1}{4.7}-\frac{1}{7.10}+...+\frac{1}{58.61}-\frac{1}{61.64}\right)\)

\(=\frac{1}{3}.\left(\frac{1}{4}-\frac{1}{61.64}\right)=\frac{1}{3}.\left(\frac{1}{4}-\frac{1}{3904}\right)=\frac{1}{3}.\frac{975}{3904}=\frac{325}{3904}\)

\(\text{Giải :}\)

\(\frac{2}{1.4.7}+\frac{2}{4.7.10}+...+\frac{2}{58.61.64}=\frac{1}{3}.\left(\frac{6}{1.4.7}+\frac{6}{4.7.10}+...+\frac{6}{58.61.64}\right)\)

\(=\frac{1}{3}.\left(\frac{7-1}{1.4.7}+\frac{10-4}{4.7.10}+...+\frac{64-58}{58.61.64}\right)\)

\(=\frac{1}{3}.\left(\frac{1}{4}-\frac{1}{4.7}+\frac{1}{4.7}-\frac{1}{7.10}+...+\frac{1}{58.61}-\frac{1}{61.64}\right)\)

\(=\frac{1}{3}.\left(\frac{1}{4}-\frac{1}{61.64}\right)=\frac{1}{3}.\left(\frac{1}{4}-\frac{1}{3904}\right)=\frac{1}{3}.\frac{975}{3904}=\frac{325}{3904}\)

\(\text{#Hok tốt!}\)

20 tháng 7 2021

ĐK : 51x \(\ge0\Rightarrow x\ge0\)

Với \(x\ge0\)thì \(x+\frac{1}{1.3}>0;x+\frac{1}{3.5}>0;...;x+\frac{1}{99.101}>0\)

Khi đó : \(\left|x+\frac{1}{1.3}\right|+\left|x+\frac{1}{3.5}\right|+\left|x+\frac{1}{5.7}\right|+...+\left|x+\frac{1}{99.101}\right|=51x\)

<=> \(x+\frac{1}{1.3}+x+\frac{1}{3.5}+x+\frac{1}{5.7}+....+x+\frac{1}{99.101}=51x\)(50 hạng tử x ở VT)

<=> \(50x+\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+...+\frac{1}{99.101}=51x\)

<=> \(x=\frac{1}{2}.\left(\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+...+\frac{1}{99.101}\right)\)

<=> \(x=\frac{1}{2}\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{99}-\frac{1}{101}\right)\)

<=> \(x=\frac{1}{2}\left(1-\frac{1}{101}\right)=\frac{50}{101}\)

Vậy x = 50/101