Solution to previous Problem of the Week


What is the remainder when f(x)=999x999+998x998+997x997+. . .+2x2+x is divided by x-1?

This is a finite polynomial of degree 999. For us to solve this problem we need to know that f(a) is the remainder when f(x) is divided by x-a. That is the famous remainder theorem.

The remainder when f(x)=999x999+998x998+997x997+. . .+2x2+x  is divided by x-1 is also f(1).

f(1)= 999(1)999+998(1)998+997(1)997+. . .+2(1)2+(1)

f(1)= 999+998+997+. . . +2+1

Using the formula for the sum of arithmetic series we have,

f(1)=\displaystyle\frac{n(a_1+a_n)}{2} where a1 and an are the first and last term respectively, n is the number of terms.



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