How to prove that this statement is true for every positive integer n?(see picture. Thanks!

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Answer 1 · 2018-02-06T18:09:30

if n^2-n+2 is pair, #(n+1)^2-(n+1)+2" is also pair

For n=1: $n^{2} - n + 2 = 2$ $n^2-n+2=2$

If $n^{2} - n + 2$ $color(purple)(n^2)-color(red)n+2$ is pair

then:

$(n^{2} + 2 n + 1) - (n - 1) + 2$ $color(purple)((n^2+2n+1))-color(red)((n-1))+2$

$n^{2} - n + 2 + 2 n + 1 + 1 + 2$ $n^2−n+2 +2n+1 +1+2$

$n^{2} - n + 2 + 2 n + 4$ $color(blue)(n^2−n+2) +2n+4$ ,

,2n+4 is obviuously pai so the thesis is proved