If {a₁},{a₂},{a₃},........,{aₙ},...... are in G.P. and {aᵢ} > 0 for each i , the

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3 and 4 .Determinants and Matrices

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If ${a_1},{a_2},{a_3},........,{a_n},......$ are in G.P. and ${a_i} > 0$ for each $i$, then the value of the determinant $\Delta = \left| {\,\begin{array}{*{20}{c}}{\log {a_n}}&{\log {a_{n + 2}}}&{\log {a_{n + 4}}}\\{\log {a_{n + 6}}}&{\log {a_{n + 8}}}&{\log {a_{n + 10}}}\\{\log {a_{n + 12}}}&{\log {a_{n + 14}}}&{\log {a_{n + 16}}}\end{array}} \right|$ is equal to

A

$1$

B

$2$

C

$0$

D

None of these

Solution

(c) If $r$ is the common ratio, then ${a_n} = {a_1}{r^{n – 1}}$ for all $n \ge 1$

$\Rightarrow \log {a_n} = \log {a_1} + (n – 1)\log r$

= $A + (n – 1)R$, where $\log {a_1} = A$ and $\log r = R$.

Thus in $\Delta $, on applying ${C_2} \to {C_2} – {C_1}$ and ${C_3} \to {C_3} – {C_2}$, we obtain ${C_2}$and ${C_3}$ are identical.

Thus $\Delta = 0$.

Standard 12

Mathematics

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