Question

用数学归纳法证明:当n>=3时,n^n+1>(n+1)^n

Answer 1

n=3

3^(3+1)=3^4=81

(3+1)^3=4^3=64

3^(3+1)>(3+1)^3成立

设n=k k^(k+1)> 成立

k*k^k>(k+1)^k

k>[(k+1)/k]^k

=(1+1/k)^k

>[1+1/(k+1)]^k

k>[1+1/(k+1)]^k

k>[(k+2)/(k+1)]^k

k(k+1)^k>(k+2)^k

k^2(k+1)^k>k(k+2)^k

(k+1)^(k+2)>(k+2)^(k+1)

(k+1)^[(k+1)+1]>[(k+1)+1]^(k+1)

n=k+1 时成立

n^n+1>(n+1)^n得证