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Math Help - What is the image of f(a,b) = ((a+1) * (a+2b)) / 2 ?

  1. #1
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    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...
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  2. #2
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    The image is \mathbb{N}. Let x be any natural number. Can you see what the image of the vector (0,x) under f is?
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  3. #3
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    Quote Originally Posted by coquitao View Post
    The image is \mathbb{N}. Let x be any natural number. Can you see what the image of the vector (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))
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  4. #4
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    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).
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