Answer to Question #297397 in Discrete Mathematics for TEE

Question #297397

USE THE METHOD OF DIRECT TO PROVE:


If a is an odd integer, then a’2 + 3a +5 is odd.


1
Expert's answer
2022-02-14T16:30:57-0500

Thus a=2c+1a = 2c +1 for some integer c,c, by definition of an odd number. Then


a2+3a+5=(2c+1)2+3(2c+1)+5a^2+3a+5=(2c+1)^2+3(2c+1)+5

=4c2+4c+1+6c+3+5=4c2+10c+9=4c^2+4c+1+6c+3+5=4c^2+10c+9

=2(2c2+5c+4)+1=2b+1,=2(2c^2+5c+4)+1=2b+1,

where b=2c2+5c+4.b=2c^2+5c+4.

Therefore by definition of an odd number, if aa is an odd integer, then a2+3a+5a^2 + 3a +5 is odd.


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