Tính nhanh 1+2+2^2+2^3+...+2^2004}-2^2005

Cái này tag tên tú hay ace cũng được mà:

Đặt+ sưả đề:

\(A=1+2+2^2+2^3+.....+2^{2004}+2^{2005}\)

\(2A=2\left(1+2+2^2+2^3+.....+2^{2004}+2^{2005}\right)\)

\(2A=2+2^2+2^3+2^4+.....+2^{2005}+2^{2006}\)

\(2A-A=\left(2+2^2+2^3+2^4+.....+2^{2005}+2^{2006}\right)-\left(1+2+2^2+2^3+.....+2^{2004}+2^{2005}\right)\)\(A=2^{2006}-1\)

Tìm chữ số tận cùng:

a;b dễ tự làm nha

c) \(19^n+5n+1890^n\)

Xét:

n lẻ:

\(\Rightarrow19^n=\overline{....9}\)

\(\Rightarrow5n=\overline{....5}\)

\(\Rightarrow1980^n=\overline{....0}\)

\(\Leftrightarrow19^n+5n+1980^n=\overline{...9}+\overline{...5}+\overline{...0}=\overline{...4}\)

Xét: n chẵn:

\(\Rightarrow19^n=\overline{....1}\)

\(\Rightarrow5n=\overline{...0}\)

\(\Rightarrow1890^n=\overline{...0}\)

\(\Leftrightarrow19^n+5n+1980^n=\overline{...1}+\overline{...0}+\overline{...0}=\overline{...1}\)

\(2^{4n}+1\)

\(4n⋮4\)

Nên ta sẽ xét những số mũ chia hết cho 4

\(2^{1.4}=2^4=\overline{...6}\)

\(2^{2.4}=2^8=\overline{...6}\)

\(2^{3.4}=2^{12}=\overline{...6}\)

\(\Rightarrow2^{4n}=\overline{...6}\)

\(\Rightarrow2^{4n}+1=\overline{...7}\)