Question

Let $a_n=5^n+7^n$, and
$\Delta =\left|\begin{array}{ccc}{a}_{n+1}& a_{n+2}& a_{n+3}\\ a_{n+4}& a_{n+5}& a_{n+6}\\ a_{n+7}& a_{n+8}& a_{n+9}\end{array}\right|$
then $\Delta$ equals

• A. $a_{n+9}-a_n$
• B. 0
• C. $a_n^2-a_{n+1}{a}_{n+9}$
• D. none of these

Answer 1

The correct option is B 0

$\Delta =\left|\begin{array}{ccc}{5}^{n+1}+7^{n+1}& 5^{n+2}+7^{n+2}& 5^{n+3}+7^{n+3}\\ 5^{n+4}+7^{n+4}& 5^{n+5}+7^{n+5}& 5^{n+6}+7^{n+6}\\ 5^{n+7}+7^{n+7}& 5^{n+8}+7^{n+8}& 5^{n+9}+7^{n+9}\end{array}\right|$

$=\left|\begin{array}{ccc}{5}^{n+1}& 5^{n+2}& 5^{n+3}\\ 5^{n+4}& 5^{n+5}& 5^{n+6}\\ 5^{n+7}& 5^{n+8}& 5^{n+9}\end{array}\right|+\left|\begin{array}{ccc}{7}^{n+1}& 7^{n+2}& 7^{n+3}\\ 7^{n+4}& 7^{n+5}& 7^{n+6}\\ 7^{n+7}& 7^{n+8}& 7^{n+9}\end{array}\right|$

$=\left(5^{n+1}\right)\left(5^{n+2}\right)\left(5^{n+3}\right)\left|\begin{array}{ccc}1& 1& 1\\ 5^3& 5^3& 5^3\\ 5^6& 5^6& 5^6\end{array}\right|+\left(7^{n+1}\right)\left(7^{n+2}\right)\left(7^{n+3}\right)\left|\begin{array}{ccc}1& 1& 1\\ 7^3& 7^3& 7^3\\ 7^6& 7^6& 7^6\end{array}\right|=0$