Question

use mathematical induction to prove that the statement is true for every positive integer n..... 2 is a factor of n^2-n+2

Answer 1

does it work for the basis statement?

Answer 2

since 1 is the first positive integer, it has to work out for n=1

1-1+2 = 2 and 2|2 is true. so now we have a basis to start from

let n=k and rewrite as:

k^2 - k + 2; now if its ture for some k, then is it true for some k+1? try it out

(k+1)^2 - (k+1) + 2 ... does this simplify to the known format?


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

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

k^2-k+2 + 2k

but (k^2-k+2)|2 by hypothesis; and 2k|2 soo what can we conclude?

Answer 3

so it is true

Answer 4

its true without using induction as well

Answer 5

\[\frac{n^2-n+2}{2}\]
\[\frac{n(n-1)+2}{2}\]
\[\frac{n(n-1)}{2}+\frac{2}{2}\]
if n is odd, then n-1 is even and is divisible by 2
if n is even, then its divisible by 2
and that 2 is divisible by 2

Answer 6

I really do appreciate your help!