Question

Tổng A = 2+2^2+2^3+2^4+2^5+2^6+2^7+2^8+2^9+2^10 có chia hết cho 3 không?

Answer 1

\(A=2+2^2+2^3+2^4+2^5+2^6+2^7+2^8+2^9+2^{10}\)

\(A=\left(2+2^2\right)+\left(2^3+2^4\right)+\left(2^5+2^6\right)+\left(2^7+2^8\right)+\left(2^9+2^{10}\right)\)

\(A=2\left(1+2\right)+2^3\left(1+2\right)+2^5\left(1+2\right)+2^7\left(1+2\right)+2^9\left(1+2\right)\)

\(A=2\cdot3+2^3\cdot3+2^5\cdot3+2^7\cdot3+2^9\cdot3\text{ }⋮\text{ }3\)

Answer 2

\(A=2+2^2+2^3+2^4+2^5+2^6+2^7+2^8+2^9+2^{10}\)

\(A=\left(2+2^2\right)+\left(2^3+2^4\right)+\left(2^5+2^6\right)+\left(2^7+2^8\right)+\left(2^9+2^{10}\right)\)

\(A=2\left(1+2\right)+2^3\left(1+2\right)+2^5\left(1+2\right)+2^7\left(1+2\right)+2^9\left(1+2\right)\)

\(A=2\cdot3+2^3\cdot3+2^5\cdot3+2^7\cdot3+2^9\cdot3\)

\(\Rightarrow\text{ }A\text{ }⋮\text{ }3\text{ }\left(\text{ ĐPCM}\right)\)