Let a , b , c , d be real numbers such that _ k=1 ^ n (a k³+b k²+c k+d )=n⁴ for every natural number n . Then | a |+| b |+| C |+| d | is equal to

Step-by-step solution

_ k=1 ^ n (a k³+b k²+c k+d )=n⁴a _ k=1 ^ n k³+b _ k=1 ^ n k²+c _ k=1 ^ n k+ _ k=1 ^ n d=n²n⁴(12-3 a)-n³(4 b+6 a)-n²(6 c+6 b+3 a)-n(6 c+2 b+12 d)=012-3 a=0, 4 b+6 a=0, 6 c+6 b+3 a=0, 6 c+2 b+12 d=0 a=4, b=-6, c=4, d=-1|a|+|b|+|c|+|d|=15