Say whether the following is true or false and support your answer by a proof: For any integer n, the number n2+n+1 is odd?
3 Answers
Explanation:
If
If
Explanation:
-
Even numbers are numbers of the form
#2k# for some integer#k# . -
Odd numbers are numbers of the form
#2k+1# for some integer#k# .
Every integer is either odd or even.
Case
If
Then:
#n^2+n+1 = (2k)^2+2k+1#
#color(white)(n^2+n+1) = 4k^2+2k+1#
#color(white)(n^2+n+1) = 2(2k^2+k)+1#
#color(white)(n^2+n+1) = 2k_1+1#
where
So
Case
If
Then:
#n^2+n+1 = (2k+1)^2+(2k+1)+1#
#color(white)(n^2+n+1) = 4k^2+2k+1+2k+1+1#
#color(white)(n^2+n+1) = 2(2k^2+2k+1)+1#
#color(white)(n^2+n+1) = 2k_2+1#
where
So
Conclusion
So regardless of whether
Is odd.
Explanation:
Note that