Question

Chứng minh rằng với mọi số nguyên n thì:

a) \(3^{n+2}-2^{n+2}+3^n-2^n\)chia hết cho 10

b) \(3^{n+3}+3^{n+1}+2^{n+3}+2^{n+2}\)chia hết cho 6

Answer 1

a) \(3^{n+2}-2^{n+2}+3^n-2^n\)

\(\Rightarrow\left(3^n\cdot3^2+3^n\right)-\left(2^n\cdot2^2+2^n\right)\)

\(\Rightarrow3^n\left(3^2+1\right)-2^n\left(2^2+1\right)\)

\(\Rightarrow3^n\cdot10-2^n\cdot5\)

\(\Rightarrow3^n\cdot10-2^{n-1}\cdot\left(2\cdot5\right)\)

\(\Rightarrow10\left(3^n-2^n\right)\) chia hết cho 10

Answer 2

b) \(3^{n+3}+3^{n+1}+2^{n+3}+2^{n+2}\)

\(\Rightarrow3^n\cdot3^3+3^n\cdot3+2^n\cdot2^3+2^n\cdot2^2\)

\(\Rightarrow3^n\left(3^3+3\right)+2^n\left(2^3+2^2\right)\)

\(\Rightarrow3^n\cdot30+2^n\cdot12\)

\(\Rightarrow3^n\cdot6\cdot5+2^n\cdot2\cdot6\)

\(\Rightarrow6\left(3^n\cdot5+2^n\cdot2\right)\) chia hết cho 6