Let a and R be the first term and common ratio of a GP. ‌∴‌‌Tp=aRp−1=x ‌Tq=aRq−1=y ‌‌ and ‌‌‌Tr=aRr−1=z ‌⇒‌‌log‌x=log‌a+(p−1)‌log‌R ‌log‌y=log‌a+(q−1)‌log‌R ‌‌ and ‌‌‌log‌z=log‌a+(r−1)‌log‌R ‌∴|
log
x
p
1
log
y
q
1
log
z
r
1
| = |
log‌a+(p−1)‌log‌R
p
1
log‌a+(q−1)‌log‌R
q
1
log‌a+(r−1)‌log‌R
r
1
| =|
log‌a
p
1
log‌a
q
1
log‌a
r
1
| + |
(p−1)‌log‌R
p
1
(q−1)‌log‌R
q
1
(r−1)‌log‌R
r
1
| =log‌a‌|
1
p
1
1
q
1
1
r
1
| + log‌R‌|
p−1
p−1
1
q−1
q−1
1
r−1
r−1
1
|‌(C2→C2−C3) ‌=0+0=0‌‌(∵‌ two columns are identical ‌)