- #1

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prove that

if "n" is an integer , then n^2-n+2 is even

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- Mathematica
- Thread starter spanker1
- Start date

- #1

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prove that

if "n" is an integer , then n^2-n+2 is even

- #2

disregardthat

Science Advisor

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Suppose n is even. What can you say about each member of n^2-n+2?

- #3

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- #4

- 25

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to prove by principle of mathemetical induction

step 1:

put n=1

1^2-1+2=1^3=1

which is false

step 1:

put n=1

1^2-1+2=1^3=1

which is false

- #5

- 52

- 0

to prove by principle of mathemetical induction

step 1:

put n=1

1^2-1+2=1^3=1

which is false

I think what is meant by n^2-n+2 is [tex]n^2 - n +2[/tex], not [tex]n^{2-n+2}[/tex] as you seem to use.

- #6

CRGreathouse

Science Advisor

Homework Helper

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prove that

if "n" is an integer , then n^2-n+2 is even

A proof by induction would take the following form:

1. For n = 1, [itex]n^2-n+2[/itex] is even.

(subproof)

2. Suppose [itex]n^2-n+2[/itex] is even. Then [itex](n+1)^2-(n+1)+2[/itex] is even.

(subproof)

3. By induction (base case 1, inductive step 2), [itex]n^2-n+2[/itex] is even for all natural numbers n.

If you can fill in the subproofs you're done.

(If you really want all integers, of course, you'll have to prove them too -- maybe by induction on [itex](-n)^2-(-n)+2.[/itex])

- #7

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prove that

if "n" is an integer , then n^2-n+2 is even

hmm...My suggestion :

N^2 - N + 2

= N(N-1) + 2

For any integer N, if N is even , (N-1) is odd therefore N(N-1) is divisible by 2 and is an even #, so , N(N-1) + 2 is even

For any integer N, if N is odd , (N-1) is even therefore N(N-1) is divisible by 2 and is an even #, so , N(N-1) + 2 is even

Is this logical?

- #8

- 840

- 14

Now, suppose that for some integer k, that [tex]k^2 -k + 2[/tex] is even. Then for the next integer k+1

[tex](k+1)^2 -(k+1) +2 = k^2 + 2k +1 - k -1 +2 = k^2 + k +2[/tex]

Now

[tex] k^2 + k + 2 = (k^2 -k + 2) + 2k[/tex]

On the RHS, the stuff in bracket is what we started with, we assumed it was even. 2k is even for all integers k. An even number plus an even number is an even number. Therefore for the next integer, it will also be even.

Therefore for all integers n, [tex]n^2 -n +2[/tex] is even

- #9

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