Câu hỏi của Trần Cao Vỹ Lượng

Question

Rút gọn biểu thức :$A=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2012}}$

Wall HaiAnh · Accepted Answer

$A=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2012}}$$\Rightarrow A=1+\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2012}}\right)$Đặt $B=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2012}}$$2B=2\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+....+\frac{1}{2^{2012}}\right)$$2B=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2011}}$$2B-B=\left(1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2012}}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2012}}\right)$$B=1-\frac{1}{2^{2012}}$$\Rightarrow A=1+\left(1-\frac{1}{2^{2012}}\right)$$\Rightarrow A=2-\frac{1}{2^{2012}}$Câu trả lời: $A=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2012}}$$\Rightarrow A=1+\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2012}}\right)$Đặt $B=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2012}}$$2B=2\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+....+\frac{1}{2^{2012}}\right)$$2B=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2011}}$$2B-B=\left(1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2012}}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2012}}\right)$$B=1-\frac{1}{2^{2012}}$$\Rightarrow A=1+\left(1-\frac{1}{2^{2012}}\right)$$\Rightarrow A=2-\frac{1}{2^{2012}}$

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