Suppose det ` [{:(sum_(k=0)^(n)k,,sum_(k=0)^(n).^nC_(k)k^2),(sum_(k=0)^(n).^nC_(k)k^k,,sum_(k=0)^(n).^nC_(k)3^(2)):}]=0` holds for some positive integer n. then `sum_(k=0)^(n)(.^nC_(k))/(k+1)` equals ............

Answer

`6.20`

Explanation

`|{:((n(n+1))/(2),n2^(n-1)+n(n-1)2^(n-2)),(n2^(n-1),):}|=0`
`n=0` or `4(n+1)-2n-n(n-1)=0`
`4n+4-2n-n^(2)+n=0`
`4n+4-2n-n^(2)+n=0`
`3n-n^(2)+4=0impliesn^(2)-3n-4=0`
`(n-4)(n+1)=0`
`n=4`
`underset(r=0)overset(4)sum(.^(4)C_(r))/(r+1)=underset(r=0)overset(4)sum(.^(5)C_(r+1))/(5)=(2^(5)-1)/(5)=(31)/(5)=6.20`