# 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 sys

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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.