the value of `Sigma_(r=2)^(n) (-2)^(r ) |{:( ""^(n-2)C_(r-2),,""^(n-2)C_(r-1),,""^(n-2)C_(r)),(-3,,1 ,,1),(2,,-1,,0):}| (n gt 2)`

Options

A. `2n -1+(-1)^(n)`
B. `2n+1+(-1)^(n-1)`
C. `2n-3+(-1)^(n)`
D. none of these

Answer

A

Explanation

Applying `c_(1) to C_(1) +2C_(2)+C_(3)` we get
`S=overset(n)underset(r=2)(Sigma)(-2)^(r ) |{:(.^(n)C_(r),,.^(n-2)C_(r-1),,.^(n-2)C_(r )),(0,,1,,1),(0,,-1,,0):}|`
`=overset(n)underset(r=2)(Sigma) (-2)^(rn)C_(r ) =overset(n)underset(r=0)(Sigma) (-2)^(r " " n)C_(r )-(.^(n)C_(0)-2.^(n)C_(1))`
`=(1-2)^(n)-(1-2n) =2n-1+(-1)^(n)`