Question

Rút gọn :

a) A = 2¹⁰⁰ - 2⁹⁹ + 2⁹⁸ - 2⁹⁷ +.....+ 2² - 2

b) B = 3¹⁰⁰ - 3⁹⁹ + 3⁹⁸ - 3⁹⁷ +.....+ 3² - 3 + 1

Answer 1

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

\(\Rightarrow2A=2^{101}-2^{100}+2^{99}-2^{98}+...+2^3-2^2\)\(\Rightarrow2A+A=\left(2^{101}-2^{100}+2^{99}-2^{98}+..+2^3-2^2\right)+\left(2^{100}-2^{99}+2^{98}-2^{97}+...+2^2-2\right)\)\(\Rightarrow3A=2^{101}-2\)

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

Answer 2

Có: \(B=3^{100}-3^{99}+3^{98}-3^{97}+...+3^2-3+1\)

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

=>\(3B+B=\left(3^{101}-3^{100}+3^{99}-3^{98}+...+3^3-3^2+3\right)+\left(3^{100}-3^{99}+3^{98}-3^{97}+...-3+1\right)\)

=>\(4B=3^{101}-3\)

=>\(B=\frac{3^{101}-3}{4}\)