Alg8.18

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If the mth,nth and pth terms of a G.P form three consecutive terms fo a geometric sequence,prove that m,n,p\, form three consecutive terms of an arithmetic sequence.

t_m=ar^{m-1},t_n=ar^{n-1},t_p=ar^{p-1}\,

Given that \frac{t_n}{t_m}=\frac{t_p}{t_n}\,

\frac{ar^{n-1}}{ar^{m-1}}=\frac{ar^{p-1}}{ar^{n-1}}\,

(r^{n-1})^2=r^{m-1}\cdot r^{p-1}\,

r^{2n-2}=r^{m+p-2}\,

Equating the indices,we get

2n-2=m+p-2,2n=m+p,n=\frac{m+p}{2}\,

Which shows that m,n,p\, are in AP.


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