The-2-nd-term-of-a-Geometric-Progresion-G-P-is-equal-to-the-8-th-term-of-an-Arithmetic-Progresion-A-P-The-first-terms-common-difference-and-common-ratio-are-all-equal-and-non-zero-Find-th

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Question Number 175829 by pete last updated on 07/Sep/22

$$\mathrm{The}{2}^{\mathrm{nd}}\mathrm{term}\mathrm{of}\mathrm{a}\mathrm{Geometric}\mathrm{Progresion}$$$$(\mathrm{G}.\mathrm{P})\mathrm{is}\mathrm{equal}\mathrm{to}\mathrm{the}{8}^{\mathrm{th}}\mathrm{term}\mathrm{of}\mathrm{an}\mathrm{Arithmetic}$$$$\mathrm{Progresion}(\mathrm{A}.\mathrm{P}).\mathrm{The}\mathrm{first}\mathrm{terms},\mathrm{common}$$$$\mathrm{difference}\mathrm{and}\mathrm{common}\mathrm{ratio}\mathrm{are}\mathrm{all}\mathrm{equal}$$$$\mathrm{and}\mathrm{non}-\mathrm{zero}.\mathrm{Find}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{first}\mathrm{five}$$$$\mathrm{terms}\mathrm{of}\mathrm{the}\mathrm{Geometric}\mathrm{Progresion}(\mathrm{G}.\mathrm{P})$$

Answered by Ar Brandon last updated on 07/Sep/22

$$\mathrm{First}\mathrm{term}=\mathrm{common}\mathrm{difference}=\mathrm{common}\mathrm{ratio}=p$$$$2^{\mathrm{nd}}\mathrm{term}\mathrm{of}GP,p\times p,\mathrm{equals}{8}^{\mathrm{th}}\mathrm{term}\mathrm{of}AP,p+7p=8p$$$$\Rightarrow p^2=8p\Rightarrow p^2-8p=0\Rightarrow p=8$$$$\Rightarrow \mathrm{Sum}\mathrm{of}GP,S=\frac{8(8^n-1)}{7}\Rightarrow S_5=\frac{8(8^5-1)}{7}=37448$$

Commented by pete last updated on 10/Sep/22

$$\mathrm{thank}\mathrm{you}\mathrm{sir}$$

Answered by Rasheed.Sindhi last updated on 08/Sep/22

$$a=d=r$$$$AP:r,2r,\dots ,7r,8r,9r,\dots ,nr$$$$GP:r,r^2,r^3,\dots ,r^n$$$$r^2=8r\Rightarrow r=8=a$$$$S_n=\frac{a(r^n-1)}{r-1}$$$$S_5=\frac{r(r^5-1)}{r-1}=\frac{8(8^5-1)}{8-1}=\frac{8(8^5-1)}{7}$$

Commented by pete last updated on 10/Sep/22

$$\mathrm{Thanks}\mathrm{sir}$$

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