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If $a_1, a_2, a_3, \ldots . . a_r$ are in GP, then prove that the determinant $\left|\begin{array}{ccc}a_{\mathrm{r}+1} & a_{\mathrm{r}+5} & \mathrm{a}_{\mathrm{r}+9} \\ \mathrm{a}_{\mathrm{r}+7} & \mathrm{a}_{\mathrm{r}+11} & \mathrm{a}_{\mathrm{r}+15} \\ \mathrm{a}_{\mathrm{r}+11} & \mathrm{a}_{\mathrm{r}+17} & \mathrm{a}_{\mathrm{r}+21}\end{array}\right|$ is independent of $\mathrm{r}$.
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1315 Upvotes
Verified Answer
We know that,
$$
\mathrm{a}_{\mathrm{r}+1}=\mathrm{AR}^{(\mathrm{r}+1)}=\mathrm{AR}^{\mathrm{r}}
$$
where $r=$ rth term of a GP, $A=$ First term of $a$ GP and $R=$ Common ratioofGP
We have, $\left|\begin{array}{lll}a_{\mathrm{r}+1} & a_{\mathrm{r}+5} & \mathrm{a}_{\mathrm{r}+9} \\ \mathrm{a}_{\mathrm{r}+7} & \mathrm{a}_{\mathrm{r}+11} & \mathrm{a}_{\mathrm{r}+15} \\ \mathrm{a}_{\mathrm{r}+11} & \mathrm{a}_{\mathrm{r}+17} & \mathrm{a}_{\mathrm{r}+21}\end{array}\right|$
$=\left|\begin{array}{ccc}\mathrm{AR}^{\mathrm{r}} & \mathrm{AR}^{\mathrm{r}+4} & \mathrm{AR^{ \textrm {r } + 8 }} \\ \mathrm{AR}^{\mathrm{I}+6} & \mathrm{AR}^{\mathrm{r}+10} & \mathrm{AR}{ }^{\mathrm{I}+14} \\ \mathrm{AR}^{\mathrm{r}+10} & \mathrm{AR}^{\mathrm{r}+16} & A R^{\mathrm{r}+20}\end{array}\right|$
$$
=\mathrm{AR}^{\mathrm{r}} \cdot \mathrm{AR}^{\mathrm{r}+6} \cdot \mathrm{AR}^{\mathrm{r}+10}\left|\begin{array}{lll}
1 & \mathrm{AR}^4 & \mathrm{AR}^8 \\
1 & \mathrm{AR}^4 & \mathrm{AR}^8 \\
1 & \mathrm{AR}^6 & \mathrm{AR}^{10}
\end{array}\right|=0
$$
$$
\mathrm{a}_{\mathrm{r}+1}=\mathrm{AR}^{(\mathrm{r}+1)}=\mathrm{AR}^{\mathrm{r}}
$$
where $r=$ rth term of a GP, $A=$ First term of $a$ GP and $R=$ Common ratioofGP
We have, $\left|\begin{array}{lll}a_{\mathrm{r}+1} & a_{\mathrm{r}+5} & \mathrm{a}_{\mathrm{r}+9} \\ \mathrm{a}_{\mathrm{r}+7} & \mathrm{a}_{\mathrm{r}+11} & \mathrm{a}_{\mathrm{r}+15} \\ \mathrm{a}_{\mathrm{r}+11} & \mathrm{a}_{\mathrm{r}+17} & \mathrm{a}_{\mathrm{r}+21}\end{array}\right|$
$=\left|\begin{array}{ccc}\mathrm{AR}^{\mathrm{r}} & \mathrm{AR}^{\mathrm{r}+4} & \mathrm{AR^{ \textrm {r } + 8 }} \\ \mathrm{AR}^{\mathrm{I}+6} & \mathrm{AR}^{\mathrm{r}+10} & \mathrm{AR}{ }^{\mathrm{I}+14} \\ \mathrm{AR}^{\mathrm{r}+10} & \mathrm{AR}^{\mathrm{r}+16} & A R^{\mathrm{r}+20}\end{array}\right|$
$$
=\mathrm{AR}^{\mathrm{r}} \cdot \mathrm{AR}^{\mathrm{r}+6} \cdot \mathrm{AR}^{\mathrm{r}+10}\left|\begin{array}{lll}
1 & \mathrm{AR}^4 & \mathrm{AR}^8 \\
1 & \mathrm{AR}^4 & \mathrm{AR}^8 \\
1 & \mathrm{AR}^6 & \mathrm{AR}^{10}
\end{array}\right|=0
$$
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