NDA Math Determinants

Concept:
Let us consider sequence a1,a2,a3...an is a G.P.
So, common ratio =r=‌

ap+1

ap

,
Where p∈ Positive Integer Logarithmic formula:
log a −log‌b=‌

log‌a

log‌b

Calculation:
Let r be the common ratio of the given GP
Now,
|

ln‌a1	ln‌a2	ln‌a3
ln‌a4	ln‌a5	ln‌a6
ln‌a7	ln‌a8	ln‌a9

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Apply, C2→C2−C1 and C3→C3−C1
=|

ln‌a1	ln‌a2−ln‌a1	ln‌a3−ln‌a2
ln‌a4	ln‌a5−ln‌a4	ln‌a6−ln‌a5
ln‌a7	ln‌a8−ln‌a7	ln‌a9−ln‌a8

|
|

ln‌a1	lnr	lnr
ln‌a4	lnr	lnr
ln‌a7	lnr	lnr

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[ Since ‌

ap+1

ap

=r and log‌ap+1−log‌ap=log‌

ap+1

ap

=log‌r ]
=0‌‌[ Since C3=C2]