Question

If $\begin{array}{c}a=1+2+4+....tonterms\\ b=1+3+9+....tonterms\\ c=1+5+25+....tonterms\end{array}$ , then $\left|\begin{array}{ccc}a& 2b& 4c\\ 2& 2& 2\\ 2^n& 3^n& 5^n\end{array}\right|$ $=$

• A. ${30}^n$
• B. ${10}^n$
• C. 0
• D. $2^n+3^n+5^n$

Answer 1

The correct option is C 0

Given, $a=1+2+4+....tonterms$
$\Rightarrow a=2^n-1$
$b=1+3+9+....tonterms$
$\Rightarrow b=\frac{3^n-1}{2}$
$c=1+5+25+....tonterms$
$\Rightarrow c=\frac{5^n-1}{4}$
Now, $\left|\begin{array}{ccc}a& 2b& 4c\\ 2& 2& 2\\ 2^n& 3^n& 5^n\end{array}\right|$
$=\left|\begin{array}{ccc}{2}^n-1& 3^n-1& 5^n-1\\ 2& 2& 2\\ 2^n& 3^n& 5^n\end{array}\right|$
$=\left|\begin{array}{ccc}{2}^n& 3^n& 5^n\\ 2& 2& 2\\ 2^n& 3^n& 5^n\end{array}\right|+\left|\begin{array}{ccc}-1& -1& -1\\ 2& 2& 2\\ 2^n& 3^n& 5^n\end{array}\right|$
$=0-2\left|\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 2^n& 3^n& 5^n\end{array}\right|$
$=0$