Given, a, b and c are pth, qth and rth terms of GP, respectively. Let a1 and k be the first term and common ratio of a GP. Then,
Tp=a1kp−1 i.e. a=a1kp−1
Tq=a1kq−1 i.e. b=a1kq−1
Tt=a1kr−1 i.e. c=a1kr−1
} .......(i)
Now, =|
log‌a
p
1
log‌b
q
1
log‌c
r
1
|=|
log(a1.kp−1)
p
1
log(a1.kq−1)
q
1
log(a1.kr−1)
r
1
| [from Eq. (i)] =|
log‌a1+log‌kp−1
p
1
log‌a1+log‌kq−1
q
1
log‌a1+log‌kr−1
r
1
|[∵log‌x‌y=log‌x+log‌y] =|
log‌a1
p
1
log‌a1
q
1
log‌a1
r
1
|+|
p−1‌log‌k
p
1
q−1‌log‌k
q
1
r−1‌log‌k
r
1
| [by property of determinant ] =log‌a1|
1
p
1
1
q
1
1
r
1
|+log‌k|
p−1
p
1
q−1
q
1
r−1
r
1
| [∵ taking log a ‌1 common from C in I determinant and log k common from C1 in II determinant ] On applying C1→C1+C3 in second determinant, we get =log‌a1(0)+log‌k|
p
p
1
q
q
1
r
r
1
| [∵C1 and C3 are identical in first determinant ] =0+log‌k(0)=0 [∵C1 and C2 are identical ]