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If a,b,c are positive and are the pth, q...

If a,b,c are positive and are the `pth, qth rth` terms respectively of a GP then `|(loga,p,1),(logb,q,1),(logc, r,1)|=`

Text Solution

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Let `a_1` is the first term of the GP and `x` is the common ratio, then we are given,
`a = a_1x^(p-1)`
`b = a_1x^(q-1)`
`c = a_1x^(r-1)`
So, with these values, our determinant becomes,
`|[log(a_1)+(p-1)logx,p,1],[log(a_1)+(q-1)logx,q,1],[log(a_1)+(r-1)logx,r,1]| `
Now, applying, `R_1->R_1-R_3` and `R_2->R_2-R_3`
`|[(p-r)logx,p-r,0],[(q-r)logx,q-r,0],[log(a_1)+(r-1)logx,r,1]| `
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