If-1-1-3-1-3-5-1-5-7-1-7-9-1-2017-2019-x-y-where-x-y-is-in-its-lower-terms-find-the-value-of-x-y-

Question Number 126963 by ZiYangLee last updated on 25/Dec/20

$$\mathrm{If}\:\frac{\mathrm{1}}{\mathrm{1}×\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{3}×\mathrm{5}}+\frac{\mathrm{1}}{\mathrm{5}×\mathrm{7}}+\frac{\mathrm{1}}{\mathrm{7}×\mathrm{9}}+\ldots+\frac{\mathrm{1}}{\mathrm{2017}×\mathrm{2019}}=\frac{{x}}{{y}} \\ $$$$\mathrm{where}\:\frac{{x}}{{y}}\:\mathrm{is}\:\mathrm{in}\:\mathrm{its}\:\mathrm{lower}\:\mathrm{terms},\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x}+{y}. \\ $$

Answered by Olaf last updated on 25/Dec/20

$$\mathrm{S}\:=\:\underset{{n}=\mathrm{0}} {\overset{\mathrm{1008}} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{3}\right)} \\ $$$$\mathrm{S}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\underset{{n}=\mathrm{0}} {\overset{\mathrm{1008}} {\sum}}\left[\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{1}}−\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{3}}\right] \\ $$$$\mathrm{S}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left[\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2019}}\right] \\ $$$$\mathrm{S}\:=\:\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{2018}}{\mathrm{2019}}\:=\:\frac{\mathrm{1009}}{\mathrm{2019}} \\ $$$${x}\:=\:\mathrm{1009},\:{y}\:=\:\mathrm{2019},\:{x}+{y}\:=\:\mathrm{3028} \\ $$

Commented by ZiYangLee last updated on 30/Dec/20

$$\mathrm{perfect}! \\ $$

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