Answer :

Given: a_{1}, a_{2}…, a_{r} are in G.P

We know that, a_{r+1} = AR^{(r+1)-1} = AR^{r} …(i)

[∵a_{n} = ar^{n-1}, where a = first term and r = common ratio]

where A = First term of given G.P

and R = common ratio of G.P

…[from(i)]

Taking AR^{r}, AR^{r+6} and AR^{r+10} common from R_{1}, R_{2} and R_{3} respectively, we get

If any two columns (or rows) of a determinant are identical (all corresponding elements are same), then the value of determinant is zero.

Here, R_{1} and R_{2} are identical.

Hence Proved

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