Question 227206
To show that is is always true, use variables and variable expressions instead of actual numbers.<br>
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".<br>
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 <b>any</b> odd integer, we have shown that this pattern is true for any pair of consecutive odd integers!