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Question 227206: Every time I add two consecutive odd numbers I get an even number which is twice the number right in between my two odd numbers. Ex. 3+5= 8 (4x2=8), this conjecture works for all consecutive odd numbers. Now, I need to be able to show that this is always true with an algebraic explanation.
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! To show that is is always true, use variables and variable expressions instead of actual numbers.
Let x = the smaller odd integer.
Now what would the next larger odd integer be? Think about consecutive odd integers. How much more is one of them than the one before it? I hope it is not hard to see that each odd integer is exactly two more than the one before it. So:
x+2 = the next odd integer
And what is the "middle" number? I hope it is not hard to see that:
x+1 = the "middle" number.
and twice the "middle" number would be:
2(x+1)
Using the Distributive Property on this we get:
2x + 2
for "twice the middle number".
Now what is the sum of our consecutive odd integers?
(x) + (x + 2)
Simplifying this we get:
2x + 2
This matches the expression we had earlier for "twice the middle number". Since "x" can be any odd integer, we have shown that this pattern is true for any pair of consecutive odd integers!
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