Question

用数学归纳法证明 7^n - 4^n - 3^n 可以被12整除.

Answer 1

设a[n] = 7^n-4^n-3^n.

n = 1时,a[1] = 0被12整除,结论成立.

假设n = k时a[k]被12整除.

a[k+1] = 7^(k+1)-4^(k+1)-3^(k+1)

= 7·7^k-4·4^k-3·3^k

= 7(a[k]+4^k+3^k)-4·4^k-3·3^k

= 7a[k]+3·4^k+4·3^k

= 7a[k]+12·(4^(k-1)+3^(k-1)).

由a[k]被12整除,得a[k+1] = 7a[k]+12·(4^(k-1)+3^(k-1))也被12整除.

即n = k+1时结论成立.

因此命题对任意正整数n成立.