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\(B=512-\dfrac{512}{2}-\dfrac{512}{2^2}-....-\dfrac{512}{2^{10}}\)
\(=512-\left(\dfrac{512}{2}+\dfrac{512}{2^2}+....+\dfrac{512}{2^{10}}\right)\)
\(=512-\left[512\left(\dfrac{1}{2}+\dfrac{1}{2^2}+....+\dfrac{1}{2^{10}}\right)\right]\)
Đặt \(A=\dfrac{1}{2}+\dfrac{1}{2^2}+....+\dfrac{1}{2^{10}}\)
\(2A=1+\dfrac{1}{2}+...+\dfrac{1}{2^9}\)
\(\Rightarrow2A-A=1-\dfrac{1}{2^{10}}\)
\(\Rightarrow A=1-\dfrac{1}{2^{10}}\)
\(\Rightarrow B=512-\left(512.A\right)=512-\left[512.\left(1-\dfrac{1}{2^{10}}\right)\right]\)
\(=512-512.\dfrac{1023}{1024}=512-\dfrac{1023}{2}=\dfrac{1}{2}\)
B=512(1-1/2-1/2^2-1/2^3-...-1/2^10
B=512*1/1024
B=1/2
B=0.5
512-\(\frac{512}{2}\)-\(\frac{512}{2^2}\)-\(\frac{512}{2^3}\)-....-\(\frac{512}{2^{10}}\)
=512-256-\(\frac{2^9}{2^2}\)-\(\frac{2^9}{2^3}\)-\(\frac{2^9}{2^4}\)-\(\frac{2^9}{2^5}\)-\(\frac{2^9}{2^6}\)-\(\frac{2^9}{2^7}\)-\(\frac{2^9}{2^8}\)-\(\frac{2^9}{2^9}\)-\(\frac{2^9}{2^{10}}\)
=512-256-128-64-32-16-8-4-2-\(\frac{1}{2}\)
=\(\frac{3}{2}\)
Đặt \(Q=512-\frac{512}{2}-\frac{512}{2^2}-...-\frac{512}{2^{10}}\)
\(=512-512\left(\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{10}}\right)\)
Đặt A là tên biểu thức trong ngoặc ta cs:
\(2A=1+\frac{1}{2}+...+\frac{1}{2^9}\)
\(2A-A=\left(1+\frac{1}{2}+...+\frac{1}{2^9}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{10}}\right)\)
\(A=1-\frac{1}{2^{10}}\)
Thay A vào Q ta được:
\(Q=512-512\left(1-\frac{1}{2^{10}}\right)=512-512+\frac{512}{2^{10}}=\frac{2^9}{2^{10}}=\frac{1}{2}\)
\(\Rightarrow\frac{M}{512}=1-\frac{1}{2}-\frac{1}{2^2}-.....-\frac{1}{2^{10}}\)
\(\Rightarrow2.\left(\frac{M}{512}\right)=2-1-\frac{1}{2}-.....-\frac{1}{2^9}\)
\(\Rightarrow2.\left(\frac{M}{512}\right)-\frac{M}{512}=\left(2-1-\frac{1}{2}-.....-\frac{1}{2^9}\right)-\left(1-\frac{1}{2}-\frac{1}{2^2}-.....-\frac{1}{2^{10}}\right)\)
\(\Rightarrow\frac{M}{512}=-\frac{1}{2^{10}}\)
\(\Rightarrow M=-\frac{1}{2}\)
\(\text{M = 512 - 512/2 - .... - 512/2^10
= 2^9 - 2^9 / 2 - 2^9/2^2 - ...2^9/2^10
= 2^9 - 2^8 - 2^7 - 2^6 -.... - 1/2
2M = 2^10 - 2^9 - 2^8 - .... - 1
2M - M = 2^10 - 2^9 - 2^8 -... -1 - 2^9 + 2^8 + 2^7 +... + 1 + 1/2
M = 2^10 - 2.2^9 + 1/2
M = 2^10 - 2^10 + 1/2}\)
\(\text{ M =}\) \(\frac{1}{2}\)
\(B=512-\dfrac{512}{2}-\dfrac{512}{2^2}-\dfrac{512}{2^3}-...-\dfrac{512}{2^{10}}\)
\(B=512-512\left(\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+..+\dfrac{1}{2^{10}}\right)\)
Đặt: \(L=\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{10}}\)
\(2L=1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^9}\)
\(2L-L=1-\dfrac{1}{2^{10}}\Leftrightarrow L=1-\dfrac{1}{2^{10}}\)
Thay Vào B
\(B=512-512\left(1-\dfrac{1}{2^{10}}\right)=512-512+\dfrac{512}{2^{10}}=\dfrac{1}{2}\)