Prove or disprove:
For any integer n, the number n² + n + 1 is odd.
Stacey Warren - Expert brainly.com
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so we just proved it
break it up into cases
Case 1: If n is even,...
Case 2: If n is odd,...
i can't find any number that disproves it...
AriPotta you proved it works for those particular values, but you didn't prove it for all integers
2^2 + 2 + 1
4 + 2 + 1 = 7 (odd)
ok i'll try using cases
Case 1: n is an even integer
Let k be any integer, so 2k is always even.
Let n = 2k
n^2 + n + 1
(2k)^2 + (2k) + 1
4k^2 + 2k + 1
2(2k^2 + k) + 1
2q + 1 ... Let q = 2k^2 + k (you can easily show that q is always an integer)
So if n = 2k (ie if n is an even integer), then n^2 + n + 1 is equivalent to 2q + 1, which is an odd integer.
I'll let you do case 2
An even + an odd is an odd . an odd + odd is an even number, and an even + even is an even number. If n is odd n^2 is odd so we have an odd + odd + 1 which is the same as odd + even so it will be odd. If n is an even n^2 is even so n^2 +n will be even. If you add 1 to an even number it is odd.
Aripotta has the right idea. Testing numbers works for odd and even in all cases.