Question

Chúng tỏ: \(A=3+3^2+3^3+...+3^{59}+3^{60}\) chia hết cho 13

 

Answer 1

ta có :

A = 3 + 3² + 3³ + ...+ 3⁵⁹ + 3⁶⁰  

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

A = 3( 1 + 3 + 3²) + 3⁴(1+3+3²) + ...+ 3⁵⁸(1+3+3² )

A = 3. 13 + 3⁴.13 + ...+ 3⁵⁸.13 

=> A chia hết cho 13 

Answer 2

Ta chú ý : \(3+3^2+3^3=3\left(1+3+9\right)=3.13\)

\(\Rightarrow A=\left(3+3^2+3^3\right)+\left(3^4+3^5+3^6\right)+..+\left(3^{58}+3^{59}+3^{60}\right)\)

\(\Leftrightarrow A=3.13+3^4.13+...+3^{58}.13\)

\(\Leftrightarrow A=13\left(3+3^4+..+3^{58}\right)⋮13\)

Vậy A chia hết cho 13

Answer 3

3mũ2