Question

Cho A=1+2+2²+2³+........+2¹⁰ So sánh A và 2¹¹

Answer 1

\(A=1+2+2^2+2^3+..........+2^{10}\)

\(\Leftrightarrow2A=2+2^2+..............+2^{11}\)

\(\Leftrightarrow2A-A=\left(2+2^2+......+2^{11}\right)-\left(1+2+....+2^{10}\right)\)

\(\Leftrightarrow A=2^{11}-2< 2^{11}\)

\(\Leftrightarrow A< 2^{11}\)

Answer 2

\(A=1+2+2^2+2^3+.....+2^{10}\)

\(\Rightarrow2A=2\left(1+2+2^2+2^3+.....+2^{10}\right)\)

\(\Rightarrow2A=2+2^2+2^3+2^4+.....+2^{11}\)

\(\Rightarrow2A-A=\left(2+2^2+2^3+2^4+.....+2^{11}\right)-\left(1+2+2^2+2^3+......+2^{10}\right)\)\(\Rightarrow A=2^{11}-1\)

\(A< 2^{11}\)

Answer 3

ta có : \(A=1+2+2^2+2^3+...+2^{10}\)

\(A=2^0+2^1+2^2+2^3+...+2^{10}\)

\(\Rightarrow2A=2.\left(2^0+2^1+2^2+2^3+...+2^{10}\right)\)

\(2A=2^1+2^2+2^3+2^4+...+2^{11}\)

\(\Rightarrow2A-A=A=\left(2^1+2^2+2^3+2^4+...+2^{11}\right)-\left(2^0+2^1+2^2+2^3+...+2^{10}\right)\)

\(A=2^{11}-2^0=2^{11}-1< 2^{11}\)

\(\Leftrightarrow A< 2^{11}\)

vậy \(A< 2^{11}\)