对任意正整数n,求证:(3n+1)(3n-1)-(3-n)(3+n)的值是10的倍数.

问题描述:

对任意正整数n,求证:(3n+1)(3n-1)-(3-n)(3+n)的值是10的倍数.

证明:原式=(3n+1)(3n-1)-(3-n)(3+n)
=9n2-1-(9-n2
=10n2-10
=10(n+1)(n-1),
∵n为正整数,
∴(n-1)(n+1)为整数,
即(3n+1)(3n-1)-(3-n)(3+n)的值是10的倍数.