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Question Number 55893 by Tawa1 last updated on 05/Mar/19

$$\mathrm{Three}\mathrm{numbers}\mathrm{are}\mathrm{in}\mathrm{G}.\mathrm{P}\mathrm{such}\mathrm{that}\mathrm{their}\mathrm{sum}\mathrm{is}\mathbf{p}\mathrm{and}\mathrm{the}$$$$\mathrm{sum}\mathrm{of}\mathrm{their}\mathrm{square}\mathrm{is}\mathbf{q}.\mathrm{Find}\mathrm{the}\mathrm{middle}\mathrm{term}\mathrm{of}\mathrm{the}\mathrm{G}.\mathrm{P}$$

Answered by kaivan.ahmadi last updated on 05/Mar/19

$$leta,b,carenumbers,so$$$$b^2=ac.$$$$a+b+c=p\Rightarrow a^2+b^2+c^2+2ab+2ac+2bc=p^2\Rightarrow$$$$q+2ab+2b^2+2bc=p^2\Rightarrow q+2b(a+b+c)=p^2$$$$q+2bp=p^2\Rightarrow b=\frac{p^2-q}{2p}$$$$$$

Commented by Tawa1 last updated on 05/Mar/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

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