Question

 

Chứng minh rằng:

a) (1+5+5²+5³+...5²⁹⁾chia hết cho 6

b) (1+3+3^2+3^3+...+3^29) chia hết cho 13

c) (1+2+2^2+2^3+...+2^120) chia hết cho 3

d) (1+2+2^2+2^3+...+2^120) chia hết cho 15

 

 

 

Answer 1

a) (1+5+5²+5³+...5²⁹⁾chia hết cho 6

Đặt (1+5+5²+5³+...5²⁹) = A

\(A=\left(1+5\right)+\left(5^2+5^3\right)+\left(5^4+5^5\right)....+\left(5^{28}+5^{29}\right)\)

\(A=\left(1+5\right)+5^2\left(5+1\right)+5^4\left(5+1\right)+...+5^{28}\left(5+1\right)\)

\(A=6+5^2.6+5^4.6+...+5^{28}.6\)

Vậy A chia hết cho 6

b) (1+3+3^2+3^3+...+3^29) chia hết cho 13

Đặt B= (1+3+3^2+3^3+...+3^29)

\(B=\left(1+3+3^2\right)+\left(3^3+3^4+3^5\right)+...+\left(3^{27}+3^{28}+3^{29}\right)\)

\(B=13+3^3\left(1+3+3^2\right)+....+3^{27}\left(1+3+3^2\right)\)

\(B=13+3^3.13+....+3^{27}.13\)

Vậy B chia hết 13

Câu c,d tương tự.Chúc bạn học tốt