- Use math induction to show that (n2 - n + 2) is divisible by 2 for all natural numbers n.

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- Jun 1st 2010, 08:40 AMkenzie103109mathematical induction
- Use math induction to show that (n2 - n + 2) is divisible by 2 for all natural numbers n.

- Jun 1st 2010, 08:56 AMnikhil
the equation is n^2-n+2

let n=1 then

1^2-1+2=2 which is divisible by 2 hence its true for n=1.

now let it be true for n=k therfor

k^2-k+2=2r

now let us consider n=k+1

(k+1)^2-(k+1)+2

=k^2+1+2k-k-1+2

=[k^2-k+2]+2k

=2r+2k

=2(r+k) which is divisible by 2

hence whenever it is true for k it is also true for k+1

hence by PMI n^2-n+2 is divisible by 2 - Jun 1st 2010, 09:01 AMPim
1. Case for 1: (1^2 - 1 + 2) = 2 is divisible by 2

2. Case for n + 1

((n+1)^2 - (n+1) + 2) =

(n^2+2n+1 - n - 1 +2) =

(n^2+n+2) =

(n^2-n+2)+2n

Assume (n^2-n+2) is divisible by two, which gives 2*(term + n), which evidently is divisible by 2.

I'm not used to proving these things, so don't use this as your proof, however, this is the general idea. - Jun 1st 2010, 04:48 PMwonderboy1953Simpler answer
n2 - n = n(n - 1)

The product of an even number with an odd number is always even as the above equation is and dividing 2 by 2 is self evident. This completes the proof.