If a1- a 2- a 3- a n - - are in G-P- then the determinant Delta - lbeginvmatrix log an -log a - n -1-log a n -2log a-n-3-log a-n-4 -log a n-5log -n-6-log a-n-7-log a-n-8endvmatrix is equal toA- 2B- 4C- 0D- 1

The correct option is C 0Since, a_1,a_2,a_3⋯ ,a_n,⋯ are in G.P. a_n=a_1 r^n 1 ⇒ log a_n=log a_1+(n 1) log r a_n+1=a_1r^n ⇒ log a_n+1=log a_1+n log r a_n+2=a_1r ...

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Byju's Answer

Standard X

Mathematics

Linear Equations in Three Variable

If a1, a 2, a...

Question

If $a_{1}, a_{2}, a_{3} \dots, a_{n}, \dots$ are in G.P., then the determinant $Δ = ∣ ∣ ∣ ∣ \begin{matrix} l o g a_{n} & l o g a_{n + 1} & l o g a_{n + 2} l o g a_{n + 3} & l o g a_{n + 4} & l o g a_{n + 5} l o g a_{n + 6} & l o g a_{n + 7} & l o g a_{n + 8} \end{matrix} ∣ ∣ ∣ ∣$ is equal to

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Solution

Since, $a_{1}, a_{2}, a_{3} \dots, a_{n}, \dots$ are in G.P.
$a_{n} = a_{1} r^{n - 1}$
$\Rightarrow l o g a_{n} = l o g a_{1} + (n - 1) l o g r$
$a_{n + 1} = a_{1} r^{n}$
$\Rightarrow l o g a_{n + 1} = l o g a_{1} + n l o g r$
$a_{n + 2} = a_{1} r^{n + 1}$
$\Rightarrow l o g a_{n + 2} = l o g a_{1} + (n + 1) l o g r$
and so on
Now, $Δ = ∣ ∣ ∣ ∣ \begin{matrix} l o g a_{n} & l o g a_{n + 1} & l o g a_{n + 2} l o g a_{n + 3} & l o g a_{n + 4} & l o g a_{n + 5} l o g a_{n + 6} & l o g a_{n + 7} & l o g a_{n + 8} \end{matrix} ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} l o g a_{1} + (n - 1) l o g r & l o g a_{1} + n l o g r & l o g a_{1} + (n + 1) l o g r l o g a_{1} + (n + 2) l o g r & l o g a_{1} + (n + 3) l o g r & l o g a_{1} + (n + 4) l o g r l o g a_{1} + (n + 5) l o g r & l o g a_{1} + (n + 6) l o g r & l o g a_{1} + (n + 7) l o g r \end{matrix} ∣ ∣ ∣ ∣ ∣$
Applying $R_{2} \to R_{2} - R_{1}$ and $R_{3} \to R_{3} - R_{1},$
$Δ = 2 ∣ ∣ ∣ ∣ ∣ \begin{matrix} l o g a_{1} + (n - 1) l o g r & l o g a_{1} + n l o g r & l o g a_{1} + (n + 1) l o g r 3 l o g r & 3 l o g r & 3 l o g r 3 l o g r & 3 l o g r & 3 l o g r \end{matrix} ∣ ∣ ∣ ∣ ∣$
=0 (since two rows are identical)

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If a1,a2,a3⋯,an,⋯ are in G.P., then the determinant Δ=∣∣ ∣∣loganlogan+1logan+2logan+3logan+4logan+5logan+6logan+7logan+8∣∣ ∣∣ is equal to