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Don't know if I got this right. Prove n^2>n+1

  • #1

Homework Statement


The principal of mathematical induction can be extended as follows. A list P(m),P(m+1)... of propositions is true provided 1)P(m) is true, 2) P(n+1) is true whenever P(n) is true and n>(or =) m

I have to use the above to prove that n^2>n+1 for n>(or equal to) 2


Homework Equations


n^2>n+1 for n>(or equal to) 2




The Attempt at a Solution



so I said m=n+1

Then since I assume that the original statement implies that I hold m constant and increase n by 1

Inductive step (n+1)^2>(n+1)
=> n+1>1 True b/c {1,2,3,...n}=N
 

Answers and Replies

  • #2
rock.freak667
Homework Helper
6,230
31

The Attempt at a Solution



so I said m=n+1

Then since I assume that the original statement implies that I hold m constant and increase n by 1

Inductive step (n+1)^2>(n+1)
=> n+1>1 True b/c {1,2,3,...n}=N
Assume true for n=k

[itex]k^2>k+1 for k \geq 2[/itex]

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

and [itex]k^2+2k+1=(k+1)^2[/itex] which is what you need on the left side. Deal with the right side now.
 
  • #3
Right side: (k+1+2k+1)=(2k+k+2)>(k+2)

Right?
 
  • #4
rock.freak667
Homework Helper
6,230
31
Right side: (k+1+2k+1)=(2k+k+2)>(k+2)

Right?
2k+k+2>k+2 => 3k>k which is true so that 2k+k+2>k+2 is true

and now you have

(k+1)^2>2k+k+2>k+2
 
  • #5
Why do you have to show that many steps?
 
  • #6
rock.freak667
Homework Helper
6,230
31
Because that is to show how P(k) => P(k+1) instead of putting n=k+1 in the formula and showing it is true.
 

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