A number theory problem
Number Theory
Level
2
If =c and a+b+c=0 where k=2n then , what is ‘b’ ? IF a≥b and 'a' and 'c' ∈ R
non negative
negative
non positive
positive
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1 solution
Since k is even we know that both a^k and b^k are non negative, so their sum, c, is also non negative. looking at a + b + c = 0 => a + b = -c, the definiton of non negative is c ≥ 0 , multiplying both sides by minus 1 gives -c ≤ 0 , using this fact with our equation gives a + b ≤ 0, if both a and b were positive their sum was also positive, but since it is non positive we can conclude at least one of them is non positive, and because b is smaller than a and a positive number cant be smaller than a non positive we conclude b is non positive.