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If mth, nth and pth terms of a G.P. form three consecutive terms of a G.P. Prove that `m ,n ,a n dp` form three consecutive terms of an arithmetic system.

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Let a be the first term and r be the common ratio of the given GP. Then,
`T_(m)=ar^((m-1)), T_(n)=ar^((n-1))` and `T_(p)=ar^((p-1))`.
Since, `T_(m), T_(n), T_(p)` are in GP, we have
`T_(n)^(2)=T_(m)xxT_(p)`
`rArr [ar^((n-1))]^(2)=[ar^((m-1))xxar^((p-1))]`
`rArr a^(2) r^((2n-1))=a^(2)r^((m+p-2))`
`rArr r^((2n-2))=r^((m+p-2))`
`rArr 2n-2=m+p-2`
`rArr 2n=m+p rArr m, n, p` are in AP.
Hence, m, n, p are three consecutive terms of an AP.

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