A Question With No Common Sense
F(x,y) is defined as follows.F(x,y)=the sum of the differences between the first x consecutive positive integers (including 0)raised to the y power.This means that if x=5 and y=2,then F(5,2)=(1-0)+(4-1)+(9-4)+(16-9)+(25-16)=25.While this may seem very easy to evaluate for all x and y, there are functions(polynomials in specific)that when added for all positive consecutive integers up to x-1 will evaluate this function.These polynomials are consistent for each y(so the polynomial is the same for all x when y=1, another polynomial is consistent for all x when y=2).Find the sum of the coefficients and the constant terms of these polynomials from y=1 to y=10.
The answer is 2036.
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1 solution
I already wrote this solution and then realized I used n instead of y.Just assume n=y.This question is asking for an algorithm to compute the difference between x^n and (x+1)^n(not x-1 because the polynomials are summed to x-1).Well if we expand (x+1)^2-x^2 we get (x+1)^2 without the x^2 term.If we repeat this for cubic it is pretty obvious that the polynomials that I have been referring to this whole time are simply (x+1)^n without the highest degree term. By binomial theorem expansions the constant terms are (n choose k for all k 0 to n)-(n choose 0).By a fairly common binomial identity we have the sum of the coefficients is just 2^n-1 for n 1 to 10.It is well known that the sum of 2^n from 0 to n is (2^(n+1))-1.Since the 2^0 is not present we have 2^(10+1)-1-2^0=2046.Finally we subtract ten because the sum of coefficients for each polynomial is 2^n-1.2046-10=2036 our answer