Question

Cho B=3+3^2+3^3+...+3^60. Chứng minh rằng 

a) B chia hết cho 4

b) B chia hết cho 13

 

Answer 1

a ) 

B=(3+3²)+(3³+3⁴)+...+(3⁵⁹+3⁶⁰)

B=3(1+3)+3³(1+3)+3⁴(1+3)+...+3⁵⁹(1+3)

4(4+3³+3⁴+...+3⁵⁹)

suy ra:4(4+3³+3⁴+...+3⁵⁹)chia hết cho 4

b )

B=(3+3²+3³)+(3⁴+3⁵+3⁶)+...+(3⁵⁸+3⁵⁹+3⁶⁰)

=3(1+3+9)+3⁴(1+3+9)+...+3⁵⁸(1+3+9)

=13.3+13.3⁴+...+13.3⁵⁸

=13.(3+3⁴+...+3⁵⁸) luôn chia hết cho 13

vậy B chia hết cho 13

Answer 2

a) \(B=3+3^2+3^3+..+3^{60}\)

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

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

\(=4\left(3+3^3+...+3^{59}\right)⋮4\)

=>đpcm

b) \(B=3+3^2+3^3+..+3^{60}\)

\(=\left(3+3^2+3^3\right)+...+\left(3^{58}+3^{59}+3^{60}\right)\)

\(=3\left(1+3+3^2\right)+...+3^{58}\left(1+3+3^2\right)=13\left(3+..+3^{58}\right)⋮13\)

=>đpcm

Answer 3

a) \(B=3+3^2+3^3+...+3^{60}\)

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

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

\(B=3.4+3^3.4+...+3^{59}.4\)

\(B=\left(3+3^3+...+3^{59}\right).4⋮4\left(đpcm\right)\)

b) \(B=3+3^2+3^3+...+3^{60}\)

\(B=\left(3+3^2+3^3\right)+...+\left(3^{58}+3^{59}+3^{60}\right)\)

\(B=3\left(1+3+3^2\right)+...+3^{58}\left(1+3+3^2\right)\)

\(B=3.13+...+3^{58}.13\)

\(B=\left(3+...+3^{58}\right).13⋮13\left(đpcm\right)\)