Question

BÀI 1: RÚT GỌN

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

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

Answer 1

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

=>3A=2^ 201 -2

=>A=\(\frac{2^{201}-2}{3}\)

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=3^ 101 -3^100+3^ 99 -3 ^98+...+3 ^3 -3 ^2+3+3 ^100 -3^ 99+3^ 98 -3^ 97+....+3 ^2 -3+1

=>4B=3 ^101+1

=>B=\(\frac{3^{101}+1}{4}\)