Let a,b,c,d be real nos. such that sigma lower limit k=1 and upper limit n(ak^3+bk^2+ck+d) = n^4 for every natural no. "n". Then |a| + |b| + |c| + |d| = ?

Question
Let a,b,c,d be real nos. such that sigma lower limit k=1 and upper limit n(ak^3+bk^2+ck+d) = n^4 for every natural no. "n". Then |a| + |b| + |c| + |d| = ?

Answer
Doing the summation:

(a/4)(n^4 + 2n^3 + n^2) + (b/6)(2n^3 + 3n^2 + n) + (c/2)(n^2 + n) + dn = n^4
n^4(a/4) + n^3(a/2 + b/3) + n^2(a/4 + b/2 + c/2) + n(c/2 + b/6 + d) = n^4
So,
a/4 = 1 , a=4
a/2 + b/3 = 0 , b=-6
a/4 + b/2 + c/2 = 0 , c=4
c/2 + b/6 + d= 0 , d=-1

Hence, |a| + |b| + |c| + |d| = 4 + 6 + 4 + 1 = 15.