I have this problem:
Find three odd consecutive numbers with the property that the product of the first one and the third one minus the product of the first one and the second is greater by eleven than the third one.
I have solved the problem with the following equation:
$(2x+1)(2x+5)-(2x+1)(2x+3)=11 + (2x+5).$
Solution is $x=7$ which gives $15,17,19$ as the requested numbers.
Now If use simpy $x, x+2, x+4$ to denote these numbers I also obtain the same solution with the equation:
Solution is $x=15$ so the numbers are $15, 17, 19$.
So my doubt is why I didnt have the need to express the numbers as a proper odd number $(2x+1)$ and it works simply with $x$.
Could be the case that there exist some other three (even) numbers that satisfy the conditions and so it's ok to use the proper expression for an odd number. Or can I always use $x, x+2, x+4, x+6$ to denote consecutive odd numbers without getting into any problems.
My question is why I was able to find the same solution using $x, x+2, x+4$ to denote the numbers, why I didn't have to use the proper $2x+1$. This tells me I can always use $x, x+2, x+4..$ to denote consecutive odd numbers to solve this kind of problems. Probably it's just a coincidence.