Question

A=2^1+2^2+2^3+2^4+2^5+2^6+....+2^60,tìm chữ số tặn cùng của A,tính A

Answer 1

Ta có: \(A=2+2^2+2^3+2^4+\cdots+2^{60}\)

\(=\left(2+2^2+2^3+2^4\right)+\left(2^5+2^6+2^7+2^8\right)+\cdots+\left(2^{57}+2^{58}+2^{59}+2^{60}\right)\)

\(=2\left(1+2+2^2+2^3\right)+2^5\left(1+2+2^2+2^3\right)+\cdots+2^{57}\left(1+2+2^2+2^3\right)\)

\(=15\left(2+2^5+\cdots+2^{57}\right)\)

\(=15\cdot2\cdot\left(1+2^4+\cdots+2^{56}\right)\)

\(=30\cdot\left(1+2^4+\cdots+2^{56}\right)\) ⋮10

=>A có chữ số tận cùng là 0

Ta có: \(A=2+2^2+2^3+2^4+\cdots+2^{60}\)

=>\(2A=2^2+2^3+2^4+2^5+\cdots+2^{61}\)

=>\(2A-A=2^2+2^3+2^4+2^5+\cdots+2^{61}-2-2^2-\cdots-2^{60}\)

=>\(A=2^{61}-2\)