If for `n in I , n > 10 ;1+(1+x)+(1+x)^2++(1+x)^n=sum_(k=0)^n a_k*x^k , x!=0` then

Options

A. `a_(n-2)=(n(n+1))/(2)`
B. `a_(9)^(2)-a_(8)^(2)=^(n+2)C_(10)('^(n+1)C_(10)-"^(n+1)C_(9))`
C. `a_(p) gt a_(p-1)` for `p lt (n)/(2)`
D. `sum_(k=0)^(n)a_(k)=2^(n+1)`

Answer

D

Explanation

`(d)` Coefficient of `x^(k)=a_(k)=^(n+1)C_(k+1)`
`a_(n-2)=^(n+1)C_(n-1)=(n(n+1))/(2)`
`a_(9)^(2)-a_(8)^(2)=^(n+2)C_(10)("^(n+1)C_(10)-^(n+1)C_(9))`
`a_(p) gt a_(p-1) implies p lt (n)/(2)`
`sum_(k=0)^(n)a_(k)=^(n+1)C_(1)+^(n+1)C_(2)+...+^(n+1)C_(n+1)=2^(n+1)-1`