Question

[do re mon]

S = 1 + 2 + 2² + 2³ + 2⁴ + 2⁵ + 2⁶ + 2⁷  Chứng minh rằng S chia hết cho 3

Answer 1

[Nguyễn Công Tỉnh]

Có S = 1 + 2 + 22 + 23 + 24 + 25 + 26 + 27

=> S = (1 + 2) + (22 + 23) + (24 + 25) + (26 + 27)

=> S = (1 + 2) + 22(1 + 2) + 24(1 + 2) + 26(1 + 2)

=> S = (1 + 2)(1 + 22 + 24 + 26)

=> S = 3(1 + 22 + 24 + 26)

=> S ⋮ 3 (đpcm)

Answer 2

[Kiệt Nguyễn]

\(S=1+2+2^2+2^3+2^4+2^5+2^6+2^7\)

\(\Leftrightarrow S=\left(1+2\right)+\left(2^2+2^3\right)+\left(2^4+2^5\right)+\left(2^6+2^7\right)\)

\(\Leftrightarrow S=2^0\left(1+2\right)+2^2\left(1+2\right)+2^4\left(1+2\right)+2^6\left(1+2\right)\)

\(\Leftrightarrow S=\left(1+2\right)\left(2^0+2^2+2^4+2^6\right)\)

\(\Leftrightarrow S=3\left(2^0+2^2+2^4+2^6\right)\)

\(\Leftrightarrow S⋮3\left(đpcm\right)\)