Question 917561: The sum of 3 consecutive odd integers is 14 more than the 2nd integer. What are the three integers?
I just don't understand how to set up an equation for this. It's very confusing to me
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! In a different type of problem,
an even integer would be represented by  , where  is an integer,
and an odd integer would be represented by  , where is an integer.
However, we can set up this type of problem using a simpler representation.
We do not worry (at first) if the integers are odd or even;
we just verify that they are what they should be at the end.
We need to choose one of those 3 odd integers as a variable.
I would use  for that second/middle odd integer,
which is the main character of this story.
(That way, 14 more than the 2nd integer is simply  ).
If is the second/middle one of the 3 consecutive odd integers,
the first of the 3 consecutive odd integers must be 2 less than ,  ,
and the third of the 3 consecutive odd integers must be 2 more than ,  .
 is the sum of the 3 consecutive odd integers,
and is 14 more than the 2nd integer.
"The sum of 3 consecutive odd integers is 14 more than the 2nd integer" translates as the equation
Solving:
Now we look at the result to see if it is an odd integer.
Since  is an odd integer, we accept that as a valid solution for the second/middle integer.
Now we need the first and the third of the 3 consecutive odd integers, which would be  and  .
The of 3 consecutive odd integers are
 ,  , and  .
NOTE:
If you do not have to "show your work",
but just need the result, found the most efficient way,
there is a mental math way.
You would be reasoning that
if the sum is 14 more than the second/middle odd integer,
the sum of the other two must be 14.
Then, since consecutive odd integers form an arithmetic progression/sequence
(like consecutive integers, or consecutive even integers),
then the sum of 2 of the first and third is twice the second one,
So 14 is twice the second/middle one.
and the second/middle one must be half of 14, which is 7.
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