jiyarajpal218 jiyarajpal218
  • 15-06-2022
  • Mathematics
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prove that n^2 +(n+1)^2 is always an odd number

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konrad509
konrad509 konrad509
  • 15-06-2022

I assume [tex]n\in\mathbb{Z}[/tex] (or at least [tex]n\in\mathbb{N}[/tex]).

[tex]n^2+(n+1)^2=n^2+n^2+2n+1=2n^2+2n+1=2(n^2+n)+1[/tex]

The above is always odd, because one of the factors of [tex]2(n^2+n)[/tex] is even, which makes the product even as well, and 1 is odd, and the sum of an even number and an odd number is an odd number.

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