The key to the consecutive integer (or consecutive even or odd integer) problems is the answer to the question: "How much more is each of these integers from the one before it?"
For consecutive integers the answer is: 1
Then, to write expressions for consecutive integers, make the variable represent the lowest integer and then, for the rest of the integers, keep adding 1:
Lowest: x
Next higher: x + 1
Next higher: x + 1 + 1 = x + 2
Next higher: x + 2 + 1 = x + 3
Next higher: x + 3 + 1 = x + 4
Next higher: x + 4 + 1 = x + 5
etc.
For consecutive even (or odd) integers the answer to the key question is: 2.
Then, to write expressions for consecutive even (or odd) integers, make the variable represent the lowest integer and then, for the rest of the integers, keep adding 2:
Lowest: x
Next higher: x + 2
Next higher: x + 2 + 2 = x + 4
Next higher: x + 4 + 2 = x + 6
Next higher: x + 6 + 2 = x + 8
Next higher: x + 8 + 2 = x + 10
etc.
In your problem you want 4 consecutive odd integers so you will use x, x+2, x+4 and x+6. It says their sum is 117. Sum means addition so:
Now we just solve this equation. Start by simplifying:
Subtract 12 from both sides:
Divide by 4:
Since this is not an integer, much less an odd integer, it means the problem, as stated is impossible. There are no 4 odd consecutive integers which add up to 117.