How can I prove this is correct??

Question

We want to know what numbers belong to the image of (a + 1)(a +2b) / 2, and I believe it's all natural numbers except those in the form of 2^n based on plugging in some numbers.

But I don't know how to start proving.

a and b are natural numbers

a cannot be 0, for natural numbers start at 1.

Answer 1

To find an image you need a function and a set. You have neither. Is this all the information given in the problem?

EDIT:
Okay, so the function is f(a,b) = (a+1)(a+2b)/2 and the set that you want the find the image of is (a,b) in N^2?

f(a,b) = (a^2+3ab+2b)/2
The number x is in the image of f when
(a^2+3ab+2b)/2 = x for some (a,b) in N^2
I'll assume here that you are including zero in the natural numbers.
Let a = 0, then you are left with
b = x
Since b is any natural number, x can be any natural number.

The expression a^2+3ab+2b is always a natural number for natural numbers a and b. Since you are dividing this by two there is the possibility that the image can also contain a natural number divided by 2. An even number divided by two will still be a natural number. An odd number on the other hand will be some number ending in .5

When a is even, a^2+3ab+2b is even. When a is odd a^2+3ab+2b is odd. So these fractional numbers will always occur when a is odd.

I'm stuck here. I don't see any other way to describe these fractional numbers other than with the original function.