Question

Assertion :

Let $\displaystyle \Delta _{n}=\begin{vmatrix}

a_{n} & a_{n+3} &a_{n+6} \\

a_{n+1} & a_{n+4} &a_{n+7} \\

a_{n+2} & a_{n+5} & a_{n+8}

\end{vmatrix}$

If $a_k>0\forall k\ge 1and{a}_1,a_2,a_3...$ are in G.P then

${\Delta }_n=0\forall n\ge 1$ Reason: If $a_1,a_2,a_3...$ are in A.P then
${\Delta }_n=0\forall n\ge 1$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Let $a_k=a{r}^{k-1}\forall k\ge 1$ then
${\Delta }_n=a_n{a}_{n+3}{a}_{n+6}\left|\begin{array}{ccc}1& 1& 1\\ r& r& r\\ r^2& r^2& r^2\end{array}\right|=0$

Next if $a_k=b+(k-1)d,$ then using $C_2\to C_3-C_{1}and{C}_2\to C_2-C_1$ we get
${\Delta }_n=\left|\begin{array}{ccc}{a}_n& 3d& 3d\\ a_{n+1}& 3d& 3d\\ a_{n+2}& 3d& 3d\end{array}\right|=0$
$[\because C_{2}and{C}_{3}areidentical]$