Question

$\Delta \left(r\right)=\left|\begin{array}{ccc}r& r^2& r^3\\ 1& 2& 4\\ 2& 1& 3\end{array}\right|$ then ${\sum }_{r=1}^5\Delta \left(r\right)$ will be _____

• A. 370
• B. -370
• C. 378
• D. 356

Answer 1

The correct option is B

-370

${\sum }_{r=1}^5\Delta \left(r\right)=\left|\begin{array}{ccc}{\sum }_{r=1}^{5}r& {\sum }_{r=1}^5{r}^2& {\sum }_{r=1}^5{r}^3\\ 1& 2& 4\\ 2& 1& 3\end{array}\right|$

$=\left|\begin{array}{ccc}\frac{r(r+1)}{2}& \frac{r(r+1)(2r+1)}{6}& \frac{r^2(r+1{)}^2}{4}\\ 1& 2& 4\\ 2& 1& 3\end{array}\right|$ where r=5

$=\left|\begin{array}{ccc}15& 55& 225\\ 1& 2& 4\\ 2& 1& 3\end{array}\right|=-370$