(急)用數學歸納法證明不等式

3^n>n^3 ,
when n = 4 , 3^4 > 4^3 is true.
Let 3^k > k^3 be true for n = k , (k > 3)
When n = k+1 :
3^(k+1)
= 3(3^k) > 3(k^3) = k^3 + k(k^2) + (k)(k)(k)
> k^3 + k(k^2) + (3)(k)(k) = k^3 + k(k^2) + (3k) + 3k(k-1)
> k^3 + k(k^2) + (3k) + 3*3(3-1) = k^3 + k(k^2) + (3k) + 18
> k^3 + 3k^2 + 3k + 18
> k^3 + 3k^2 + 3k + 1 = (k+1)^3

由數學歸納法 , 當 k>3 為真 , k+1 亦為真。

得證。