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Question Number 6449 by Temp last updated on 27/Jun/16

$$x=0.112123123412345123456...$$$$\mathrm{decimal}\mathrm{pattern}:1,12,123,...$$$$\mathrm{Can}\mathrm{this}\mathrm{be}\mathrm{represented}\mathrm{as}\mathrm{a}\mathrm{fraction}?$$$$\mathrm{or}\mathrm{is}\mathrm{this}\mathrm{number}\mathrm{trancendental}?$$

Commented by prakash jain last updated on 28/Jun/16

$$123456\mathrm{may}\mathrm{have}\mathrm{some}\mathrm{inputs}\mathrm{on}\mathrm{this}$$$$\mathrm{sequence}.$$

Commented by prakash jain last updated on 27/Jun/16

$$\mathrm{What}\mathrm{comes}\mathrm{after}$$$$12345689$$$$12345678910?$$$$\mathrm{If}\mathrm{the}\mathrm{pattern}\mathrm{is}\mathrm{non}-\mathrm{repeating}\mathrm{then}\mathrm{it}$$$$\mathrm{will}\mathrm{be}\mathrm{irrational}.$$$$\mathrm{It}\mathrm{is}\mathrm{generally}\mathrm{very}\mathrm{hard}\mathrm{to}\mathrm{prove}\mathrm{if}\mathrm{a}$$$$\mathrm{number}\mathrm{is}\mathrm{trancendental}.$$

Commented by Temp last updated on 28/Jun/16

$$12345678910,1234567891011,etc.$$$$\mathrm{and}\mathrm{I}\mathrm{see}$$

Commented by nburiburu last updated on 28/Jun/16

$$thisnumberisgeneralizedas$$$$x=\underset{n=0}{\sum ^{\mathrm{\infty }}}\left(\frac{n-\frac{\lfloor \frac{\sqrt{8n+1}-1}{2}\rfloor (\lfloor \frac{\sqrt{8n+1}-1}{2}\rfloor +1)}{2}+1}{{10}^{n(n+1)/2}}\right)$$$$(Smadaranchecrecendosequence/$$$$10tothesumfrom1ton)$$$$$$$$Itisnotarationalnumber.$$$$$$

Commented by prakash jain last updated on 28/Jun/16

$$\mathrm{This}\mathrm{is}\mathrm{good}\mathrm{information}.\mathrm{Thanks}.$$$$\mathrm{I}\mathrm{will}\mathrm{look}\mathrm{up}\mathrm{derivation}\mathrm{of}\mathrm{the}\mathrm{formula}.$$

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