# Problem: Learning to recognize patterns which suggest general truths. For example looking at the statements. 3+5=8, 9+5=14, 11+17=28 you might guess that the sum of two odd numbers is an even number. Can you think of two odd numbers whose sum is an odd number? Consider these statements: If 3+5=9, 5+7+11=23, 9+13+21=43 a.) Complete this generalization If a,b and c are three odd numbers, then a+b+c is ____________ b.) Can you establish the truth of this generalization? Please help and show your solution please... :D

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by georgiabocanegra

2015-06-06T20:57:52+08:00

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For the problem above, no I cannot think of two numbers whose sum is an odd number.

Consider these statements:
If 3+5=8   5+7+11 = 23    9+13+21=43
a) If a,b,c are three odd numbers then a+b+c is an odd number as well.

b) An odd number is represented by 2m+1 with m∈Z

So we let:
a = 2x+1
b = 2y+1
c = 2z + 1

with x,y,z∈Z

To show that the statement above about the sum of two number being even:
a + b = (2x+1) + (2y+1) = 2x+2y+2 = 2(x+y+1)
Therefore it is TRUE

but is it true that 3 odd numbers have a sum that is odd?
a + b + c = (2x+1) + (2y+1) + (2z+1) = 2x + 2y + 2z + 3= = 2(x+y+z+1) + 1
We let x+y+z+1=m, since x,y,z are integers then m is an integer as well:
a+b+c = 2m + 1
This has the same form as above
Therefore the sum of three odd numbers is odd