Induction proof of ( n+1/n)^n <= n for n>=3

[joserduarte]

a) Proof using induction ( n+1/n)^n <= n for n>=3.
b) Using item a proof 1, sqrt(2), (3) ^(1/3), (4)^(1/4), ... Is decreasing from the third term.
Thanks

 

[Deleted member 4993]

a) Proof using induction ( n+1/n)^n <= n for n>=3.
b) Using item a proof 1, sqrt(2), (3) ^(1/3), (4)^(1/4), ... Is decreasing from the third term.
Thanks

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[Steven G]

a) Proof using induction ( n+1/n)^n <= n for n>=3.
b) Using item a proof 1, sqrt(2), (3) ^(1/3), (4)^(1/4), ... Is decreasing from the third term.
Thanks

Hi, welcome to the math help forum. I like the problems you posted. I am however confused as to why you posted them. Do you have a question related to them? I asked this because you did not ask any question.

 

[lookagain]

a) Proof using induction ( n+1/n)^n <= n for n>=3

That means \(\displaystyle \ \ \bigg(n \ + \ \dfrac{1}{n}\bigg)^n \ \le \ n \ \ \ for \ \ n \ \ge \ 3.\)


That's not true.


Did you mean \(\displaystyle \ \ \bigg( \dfrac{n + 1}{n}\bigg)^n \ \le \ n \ \ \ for \ \ n \ \ge \ 3 \ ?\)