# Thread: What is the image of f(a,b) = ((a+1) * (a+2b)) / 2 ?

1. ## What is the image of f(a,b) = ((a+1) * (a+2b)) / 2 ?

Hi,
I am looking for the image of the following function:

f(a,b) = ((a+1) * (a+2b)) / 2

a and b are natural numbers and the image is also contained in natural numbers.

For the solution of the problem, I plugged in even and/or odd numbers for a and b and tried to see a pattern. But, I could not see a distinct pattern. When the numbers are small, it seems that we can get almost as many odd numbers as even numbers as the output, but when a and b get larger and have lots of zeros, even results seemed to come out more.

I would really appreciate, if you could give your thoughts on this...

2. The image is $\displaystyle \mathbb{N}$. Let x be any natural number. Can you see what the image of the vector $\displaystyle (0,x)$ under f is?

3. Originally Posted by coquitao
The image is $\displaystyle \mathbb{N}$. Let x be any natural number. Can you see what the image of the vector $\displaystyle (0,x)$ under f is?
I see your point, but how can you compare your example to the formula above? I do not see a clear connection between them. ((O,x) vs. the formula for f(a,b))

4. Well, f(0,x) = (0+1)*(0+2x)/2=x which implies that every natural number x can be found in the range of x (as long as you are assuming that the set of natural numbers starts with 0).