Question

chứng minh rằng :với mọi số nguyên dương n thì :\(3^{n+2}-2^{n+2}+3^n-2^n\)   chia hết cho 10

Answer 1

Ta có: \(3^{n+2}-2^{n+2}+3^n-2^n=3^{n+2}+3^n-\left(2^{n+2}+2^n\right)\)

Thấy: \(3^{n+2}+3^n=3^n.2^2+3^n=9.3^n+3^n=3^n.\left(9+1\right)=3^n.10\)

\(\Rightarrow3^{n+2}+3^n⋮10\)\(\left(1\right)\)

\(2^{n+2}+2^n=4.2^n+2^n==2^n\left(4+1\right)=2^n.5=2.2^{n-1}.5=10.2^{n-1}\)

\(\Rightarrow2^{n+2}+2^n⋮10\)\(\left(2\right)\)

Từ (1) và (2) \(\Rightarrow3^{n+2}+2^n-\left(2^{n+2}+2^n\right)⋮10\Rightarrow3^{n+2}-2^{n+2}+3^n-2^n⋮10\) (đpcm)

k!