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Question Number 46921 by 786786AM last updated on 02/Nov/18

$$\mathrm{The}\mathrm{sum}\mathrm{of}\mathrm{first}\mathrm{two}\mathrm{terms}\mathrm{of}\mathrm{an}\mathrm{infinite}$$$$\mathrm{GP}\mathrm{is}1\mathrm{and}\mathrm{every}\mathrm{term}\mathrm{is}\mathrm{twice}\mathrm{the}\mathrm{sum}$$$$\mathrm{of}\mathrm{the}\mathrm{successive}\mathrm{terms}.\mathrm{Its}\mathrm{first}\mathrm{term}\mathrm{is}$$

Answered by peter frank last updated on 02/Nov/18

$$\mathrm{a}+\mathrm{ar}=1.....\left(1\right)$$$$\mathrm{a}=2(\mathrm{ar}+{\mathrm{ar}}^2+{\mathrm{ar}}^3+...)...\left(2\right)$$$$\mathrm{from}1$$$$\mathrm{a}=\frac{1}{1+\mathrm{r}}$$$$\mathrm{from}2$$$$\mathrm{a}=\frac{2\mathrm{ar}}{1-\mathrm{r}}$$$$1-\mathrm{r}=2\mathrm{r}$$$$\mathrm{r}=\frac{1}{3}$$$$\mathrm{substitute}\mathrm{r}\mathrm{in}1$$$$\mathrm{a}=\frac{3}{4}$$$$$$

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