# P.o.t.W. #11 Solution

SOLUTIONLet $${u_1},\;{u_1}r$$ and $${u_1}{r^2}$$ be the three consecutive terms of the geometric sequence.If $${u_1}$$ and $${u_1}r$$ are the first and fourth terms, respectively, of an arithmetic sequence then $${u_1}r - {u_1} = 3d$$; and if $${u_1}r$$ and $${u_1}{r^2}$$ are the first and eighth terms, respectively, then $${u_1}{r^2} - {u_1} = 7d$$.Solving for $$d$$ in each of these equations gives:$${u_1}r - {u_1} = 3d\;\;\; \Rightarrow \;\;\;d = \frac{1}{3}\left( {{u_1}r - {u_1}} \right)$$;and $${u_1}{r^2} - {u_1} = 7d\;\;\; \Rightarrow \;\;\;d = \frac{1}{7}\left( {{u_1}{r^2} - {u_1}} \right)$$$$d = \frac{1}{3}\left( {{u_1}r - {u_1}} \right) = \frac{1}{7}\left( {{u_1}{r^2} - {u_1}} \right)$$

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