Question

Cho:

A=1+3+3²+3³+.....+3¹⁹⁹⁹+3²⁰⁰⁰

Chứng minh rằng A đã chia hết cho 13

Answer 1

A = 1+ 3 + 3² + 3³ + .... + 3 ¹⁹⁹⁹ + 3²⁰⁰⁰ 

3A= \(3\times\left(1+3+3^2+3^3+...+3^{1999}+3^{2000}\right)\)

\(3A=3+3^2+3^3+3^4+...+3^{2000}+3^{2001}\)

\(3A=3\times\left(1+3+3^2\right)+3^4\times\left(1+3+3^2\right)+...+3^{1999}\times\left(1+3+3^2\right)\)

\(3A=3\times13+3^4\times13+...+3^{1999}\times13\)

\(3A=13\times\left(3+3^4+...+3^{1999}\right)\Rightarrow3A\)chia hết cho 13 \(\Rightarrow A\)chia hết cho 13