Question

cho C=1/3+1/3^2+1/3^3+...+1/3^99 CMR C<1/2

 

Answer 1

A=2^100-2^99+2^98-2^97+...+2^2-2

=>2A=2¹⁰¹-2¹⁰⁰+2⁹⁹-2⁹⁸+...+2³-2²

=>A+2A=2¹⁰⁰-2⁹⁹+2⁹⁸-2⁹⁷+...+2²-2+2¹⁰¹-2¹⁰⁰+2⁹⁹-2⁹⁸+...+2³-2²

=>3A=2¹⁰¹-2

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

Answer 2

Bạn tham khảo nhé:

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

\(\Rightarrow2A=2^{101}-2^{100}+2^{99}-2^{98}+...+2^3-2^2\)

\(\Rightarrow2A+A=2^{101}-2^{100}+...+2^3-2^2+2^{100}-2^{99}+...+2^2-2\)

\(\Rightarrow3A=2^{101}-2\)

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

 Chúc bạn học tốt!