Question

rút gọn biểu thức : A = 2^100 - 2^99 + 2^98 - 2^97 +....+2^2 -2

Answer 1

Ta có:

\(A=2^{100}-2^{99}+2^{98}-2^{97}+...+2^2-2\)

\(\Rightarrow A=\left(-2\right)^{100}+\left(-2\right)^{99}+\left(-2\right)^{98}+\left(-2\right)^{97}+...+\left(-2\right)^2+\left(-2\right)\)\(\Rightarrow-2A=\)\(\left(-2\right)^{101}+\left(-2\right)^{100}+\left(-2\right)^{99}+\left(-2\right)^{98}+...+\left(-2\right)^3+\left(-2\right)^2\)

\(\Rightarrow-2A-A=\left(-2\right)^{101}-\left(-2\right)\)

\(\Rightarrow-3A=\left(-2\right)^{101}+2\)

\(\Rightarrow A=\frac{2-2^{101}}{-3}\)

Answer 2

\(A=2^{100}-2^{99}+2^{98}-2^{97}+.....+2^2-2 \)

\(2A=2^{101}-2^{100}+2^{99}-2^{98}+......+2^3-2^2\)\(2A+A=2^{101}-2^{100}+2^{99}-2^{98}+.....+2^3-2^2+2^{100}-2^{99}+2^{98}-2^{97}+.....+2^2-2\)\(2A+A=2^{101}-2\)

\(3A=2^{101}-2\)