Show that the products of the corresponding terms of the sequences a, `a r ,a r^2,.......,a r^(n-1)`and `A ,A R ,A R^2,......,A R^(n-1),`form a G.P, and find the common ratio.

Explanation

`1^(st)` sequence is `a, ar, ar^2,.....ar^(n-1)`

`2^(nd)` sequence is `A, AR, AR^2,.....AR^(n-1)`




Let `P=` Product of corresponding terms of `1^(st)` and `2^(nd)` sequence

` P= axxA, ar xx AR, ar^2 xx AR^2,......, ar^(n-1)xxAR^(n-1)`

`P=aA,arAR,ar^2AR^2,.....aA(r^(n-1)R^(n-1))`

Now, in `aA,arAR,ar^2AR^2,.....aA(r^(n-1)R^(n-1))`

`(arAR)/(aA)=rR`

& `(ar^2AR^2)/(arAR)=Rr`

Thus,`(Second term)/(First term)=(Third term)/(Second term)`, i.e. common ratio is same

Thus, it is a G.P.

Common ratio=`r=(Second term)/(First term)`

`r=(arAR)/(aA)`

`r=rR`