Question

Tính: \(2^{100}-2^{99}-2^{98}-2^{97}-..-2-1\)

HELP ME!

Answer 1

Đặt \(A=2^{100}-2^{99}-2^{98}-2^{97}-\cdot\cdot\cdot-2-1\)

\(=-\left(1+2+\cdot\cdot\cdot+2^{99}+2^{100}\right)\)

Đặt \(B=1+2+\cdot\cdot\cdot+2^{99}+2^{100}\)

\(2B=2+2^2+\cdot\cdot\cdot+2^{100}+2^{101}\)

\(2B-B=2+2^2+\cdot\cdot\cdot+2^{100}+2^{101}-\left(1+2+\cdot\cdot\cdot+2^{99}+2^{100}\right)\)

\(B=2^{101}-1\)

Thay \(B=2^{101}-1\) vào \(A\), ta được:

\(A=-\left(2^{101}-1\right)\)

\(=1-2^{101}\)

#\(Toru\)