Question

The value of $\sum _{r=2}^n(-2{)}^r\left|\begin{array}{ccc}{}^{n-2}{C}_r& {}^{n-2}{C}_{r-1}& {}^{n-2}{C}_{r-2}\\ -3& 1& 1\\ 2& -1& 0\end{array}\right|,(n>2)$ is

• A. $2n-1+(-1{)}^n$
• B. $2n-3+(-1{)}^n$
• C. $2n+1+(-1{)}^{n-1}$
• D. $2n+3+(-1{)}^n$

Answer 1

The correct option is A $2n-1+(-1{)}^n$

Applying, $C_1\to C_1+2C_2+C_3$
we get,
$S=\sum _{r=2}^n(-2{)}^r\left|\begin{array}{ccc}{}^n{C}_r& {}^{n-2}{C}_{r-1}& {}^{n-2}{C}_{r-2}\\ 0& 1& 1\\ 0& -1& 0\end{array}\right|=\sum _{r=2}^n(-2{)}^r\cdot {}^n{C}_r=\sum _{r=0}^n(-2{)}^r\cdot {}^n{C}_r-({}^n{C}_0-2\cdot {}^n{C}_1)=(1-2{)}^n-(1-2n)(\because (1-x{)}^n=\sum _{r=0}^n(-x{)}^r\cdot {}^n{C}_r)=2n-1+(-1{)}^n$