Question

$\mathrm{If}{D}_r=\left(\begin{array}{ccc}{2}^{r-1}& 2.3^{r-1}& 4.5^{r-1}\\ \alpha & \beta & \gamma \\ 2^n-1& 3^n-1& 5^n-1\end{array}\right|,\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}{\sum }_{r=1}^n{D}_r\mathrm{is}$

• A. $0$
• B. $\alpha \beta \gamma$
• C. $\alpha +\beta +\gamma$
• D. $\alpha -2^n+\beta -3^n+\gamma -4^n$

Answer 1

The correct option is A

$0$

${\sum }_{r=1}^n{2}^{r-1}=1+2+2^2+...+2^{n-1}=\frac{2^n-1}{2-1}=2^n-1{\sum }_{r=1}^{n}2.3^{r-1}=\frac{2(3^n-1)}{3-1}=3^n-1{\sum }_{r=1}^{n}4.5^{r-1}=\frac{4(5^n-1)}{5-1}=5^n-1\Rightarrow {\sum }_{r=1}^n{D}_r=\left(\begin{array}{ccc}{2}^n-1& 3^n-1& 5^n-1\\ \alpha & \beta & \gamma \\ 2^n-1& 3^n-1& 5^n-1\end{array}\right|=0$