Question

\(A=3+3^2+3^3+3^4+...+3^{49}+3^{50}\)

chứng minh rằng A chia hết cho 4

Answer 1

A=3+3^{2 +}3³+3⁴+...+3⁴⁹+3⁵⁰

=(3+3²)+(3²+3³)+...(3⁴⁹⁺3⁵⁰)

=3.(1+3)+5².(1+3)+.....+3⁴⁹+(1+3)

=3.4+3³.4+...+3⁴⁹.4

=4.(3+3³+...+3⁴⁹)chia hết cho 4

=> A chia hết cho 4

Answer 2

\(A=3+3^2+3^3+...+3^{50}\)

Ta có : \(3+3^2=3.1+3.3=3.\left(1+3\right)=3.4⋮4\)

\(3^3+3^4=3^2.1+3^2.3=3^2.\left(1+3\right)=3^2.4⋮4\)

......... ..... .......... .........

\(3^{49}+3^{50}=3^{49}.1+3^{49}.3=3^{49}.\left(1+3\right)=3^{49}.4⋮4\)

\(\Rightarrow\left\{3+3^2+3^3+3^4...+3^{49}+3^{50}\right\}⋮4\)

\(\Rightarrow A⋮4\)