ICSE Class 10 Answered

In this GP you if first term is x and common ratio (ratio of a term to its preceding term) is y

Then

xy^(p-1)=a

Or log {xy^(p-1)}=Log a

Or log x + log {y^(p-1)}=log a

Or log x + (p-1)log y=log a………(1)

xy^(q-1)=b

Or log x+(q-1)log y= log b……….. (2)

xy^(r-1)=c

Or log x+(r-1)log y=log c………..(3)

From (1), (2), (3) The expression

(q-r)log a+(r-p)log b+ (p-q)log c becomes

(q-r){log x + (p-1) log y} +(r-p){log x +(q-1)log y} +(p-q){log x +(r-1)log y}

= (q-r+r-p+p-q)log x + {(q-r)(p-1)+(r-p)(q-1)+(p-q)(r-1)}log y

=0+{pq-pr-q+r+qr-pq-r+p+pr-qr-p+q}log y

=0