Ta có:
\(P=512-\frac{512}{2}-\frac{512}{2^2}-\frac{512}{2^3}-...-\frac{512}{2^{10}}\)
\(\Rightarrow P=512-\left(\frac{512}{2}+\frac{512}{2^2}+\frac{512}{2^3}+...+\frac{512}{2^{10}}\right)\)
Đặt \(A=\frac{512}{2}+\frac{512}{2^2}+\frac{512}{2^3}+...+\frac{512}{2^{10}}\)
\(\Rightarrow2A=512+\frac{512}{2}+\frac{512}{2^2}+...+\frac{512}{2^9}\)
\(\Rightarrow2A-A=512-\frac{512}{2^{10}}\)
\(\Rightarrow A=512-\frac{512}{2^{10}}\)
\(\Rightarrow P=512-A=512-\left(512-\frac{512}{2^{10}}\right)=\frac{512}{2^{10}}=\frac{1}{2}\)
\(P=512-\frac{512}{2}-\frac{512}{2^2}-\frac{512}{2^3}-...-\frac{512}{2^{10}}\)
\(\Rightarrow P=2^9-\frac{2^9}{2}-\frac{2^9}{2^2}-\frac{2^9}{2^3}-...-\frac{2^9}{2^{10}}\)
\(\Rightarrow P=2^9-2^8-2^7-2^6-...-\frac{1}{2}\)
\(\Rightarrow2P=2^{10}-2^9-2^8-2^7-...-1\)
\(\Rightarrow2P-P=2^{10}-2^9-2^8-2^7-...-1-\left(2^9-2^8-2^7-2^6-...-\frac{1}{2}\right)\)
\(\Rightarrow2P-P=2^{10}-2^9-2^8-2^7-...-1-2^9+2^8+2^7+2^6+...+\frac{1}{2}\)
\(\Rightarrow P=2^{10}-2^9-2^9+\frac{1}{2}\)
\(\Rightarrow P=2^{10}-2.2^9+\frac{1}{2}\)
\(\Rightarrow P=2^{10}-2^{10}+\frac{1}{2}\)
\(\Rightarrow P=0+\frac{1}{2}\)
\(\Rightarrow P=\frac{1}{2}.\)
Chúc bạn học tốt!
