If f: N Z is defined by f(n) = vmatrix n & -1 & -5 -2n^2 & 3(2k+1) & 2k+1 -3n^3 & 3k(2k+1) & 3k(k+2)+1 vmatrix , k N , and _ n=1 ^ k f(n) = 98 , then k is equal to :

Step-by-step solution

Let S = _ n=1 ^ k f(n) , where f(n) = vmatrix n & -1 & -5 -2n^2 & 3(2k+1) & 2k+1 -3n^3 & 3k(2k+1) & 3k(k+2)+1 vmatrix Since only the first column depends on n , the summation can be taken inside that column: S = vmatrix _ n=1 ^ k n & -1 & -5 _ n=1 ^ k (-2n^2) & 3(2k+1) & 2k+1 _ n=1 ^ k (-3n^3) & 3k(2k+1) & 3k(k+2)+1 vmatrix Using standard summation formulas: _ n=1 ^ k n = k(k+1) 2 _ n=1 ^ k (-2n^2) = - k(k+1)(2k+1) 3 _ n=1 ^ k (-3n^3) = - 3k^2(k+1)^2 4 Substituting: S = vmatrix k(k+1) 2 & -1 & -5 - k(k+1)(2k+1) 3 &