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(C) Let $r$ be the common ratio of the $G$.$P$. Then ${a_n} = {a_1}{r^{n-1}}$.Taking logarithm on both sides,we get $\log {a_n} = \log {a_1} + (n-1)\l

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(C) Let $r$ be the common ratio of the $G$.$P$. Then ${a_n} = {a_1}{r^{n-1}}$.Taking logarithm on both sides,we get $\log {a_n} = \log {a_1} + (n-1)\log r$.Let $A = \log {a_1}$ and $R = \log r$. Then $\log {a_n} = A + (n-1)R$.Now,the terms in the determinant are of the form $\log {a_{n+k}} = A + (n+k-1)R$.Applying column operations ${C_2} 	o {C_2} - {C_1}$ and ${C_3} 	o {C_3} - {C_2}$:${C_2} - {C_1}$ results in entries: $(n+2-1)R - (n-1)R = 2R$,$(n+8-1)R - (n+6-1)R = 2R$,and $(n+14-1)R - (n+12-1)R = 2R$.${C_3} - {C_2}$ results in entries: $(n+4-1)R - (n+2-1)R = 2R$,$(n+10-1)R - (n+8-1)R = 2R$,and $(n+16-1)R - (n+14-1)R = 2R$.Since columns ${C_2}$ and ${C_3}$ become identical (both consist of $2R$),the value of the determinant $\Delta = 0$.

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