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Question Number 135609 by physicstutes last updated on 14/Mar/21

express the number (3/(13)) as a decimal ane state the value of the 300^(th) value

$$\mathrm{express}\:\mathrm{the}\:\mathrm{number}\:\frac{\mathrm{3}}{\mathrm{13}}\:\mathrm{as}\:\mathrm{a}\:\mathrm{decimal}\:\mathrm{ane}\:\mathrm{state}\:\mathrm{the} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{300}^{\mathrm{th}} \:\mathrm{value} \\ $$

Answered by Olaf last updated on 14/Mar/21

(3/(13)) = 0,230769^(−) we verify that 230769Σ_(n=1) ^∞ 10^(−6n) = 230769×10^(−6) (1/(1−10^(−6) )) = ((230769)/(10^6 −1)) = ((230769)/(999999)) = ((3×76923)/(13×76923)) = (3/(13)) 300 = 6×50 the 300th decimal is the same than the 6th decimal : 9

$$\frac{\mathrm{3}}{\mathrm{13}}\:=\:\mathrm{0},\overline {\mathrm{230769}} \\ $$$$\mathrm{we}\:\mathrm{verify}\:\mathrm{that}\:\mathrm{230769}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\mathrm{10}^{−\mathrm{6}{n}} \\ $$$$=\:\mathrm{230769}×\mathrm{10}^{−\mathrm{6}} \frac{\mathrm{1}}{\mathrm{1}−\mathrm{10}^{−\mathrm{6}} }\:=\:\frac{\mathrm{230769}}{\mathrm{10}^{\mathrm{6}} −\mathrm{1}} \\ $$$$=\:\frac{\mathrm{230769}}{\mathrm{999999}}\:=\:\frac{\mathrm{3}×\mathrm{76923}}{\mathrm{13}×\mathrm{76923}}\:=\:\frac{\mathrm{3}}{\mathrm{13}} \\ $$$$\mathrm{300}\:=\:\mathrm{6}×\mathrm{50} \\ $$$$\mathrm{the}\:\mathrm{300th}\:\mathrm{decimal}\:\mathrm{is}\:\mathrm{the}\:\mathrm{same}\:\mathrm{than} \\ $$$$\mathrm{the}\:\mathrm{6th}\:\mathrm{decimal}\::\:\mathrm{9} \\ $$

Commented by physicstutes last updated on 14/Mar/21

thanks a lot sir

$$\mathrm{thanks}\:\mathrm{a}\:\mathrm{lot}\:\mathrm{sir} \\ $$

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