若n是整数,是说明3^n+3-4^n+1-2^2n若n是正整数,试说明3^n+3-4^n+1+3^n+1-2^2n能被10整除

Question

若n是整数,是说明3^n+3-4^n+1-2^2n
若n是正整数,试说明3^n+3-4^n+1+3^n+1-2^2n能被10整除

Answer 1

3^(n+3)-4^(n+1)+3^(n+1)-2^(2n)
=3^(n+3)-2^(2n+2)+3^(n+1)-2^(2n)
=[3^(n+3)+3^(n+1)]-[2^(2n+2)+2^(2n)]
=3^(n+1)(3^2+1)-2^(2n-1)(2^3+2)
=10×3^(n+1)-10×2^(2n-1)
=10×[3^(n+1)-2^(2n-1)]
包含因数10,因此3^(n+3)-4^(n+1)+3^(n+1)-2^(2n)能被10整除.