Question

chứng tỏ tổng \(2+2^2+2^3+2^4+...+2^{59}+2^{60}\)chia hết cho 3.

Answer 1

\(2+2^2+2^3+....+2^{59}+2^{60}\)

\(=\left(2+2^2\right)+\left(2^3+2^4\right)+...+\left(2^{59}+2^{60}\right)\)

\(=2\left(1+2\right)+2^3\left(1+2\right)+.....+2^{59}\left(1+2\right)\)

\(=2.3+2^3.3+....+2^{59}.3\)

\(=3\left(2+2^3+...+2^{59}\right)\)

Vì có cơ số là 3 nên \(=3\left(2+2^3+...+2^{59}\right)\)

Vậy : \(2+2^2+2^3+....+2^{59}+2^{60}\)