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Question Number 100464 by mathocean1 last updated on 26/Jun/20

$${A}\:{car}\:{is}\:{driving}.\:{It}\:{most}\:{arrive} \\$$$${at}\:{destination}\:{situated}\:{at}\:\mathrm{2046}\:{m}. \\$$$${It}\:{has}\:{travelled}\:\mathrm{2}{m}\:{the}\:{first}\:{day},\: \\$$$$\mathrm{4}\:{m}\:{the}\:{second},\:\mathrm{8}{m}\:{the}\:{thirst},\:\mathrm{16} \\$$$${the}\:{fourth}\:{day}\:... \\$$$${How}\:{many}\:{days}\:{is}\:{necessary}\:\:{to}\: \\$$$${arrive}\:{at}\:{destination}? \\$$

Commented by Dwaipayan Shikari last updated on 26/Jun/20

$$\mathrm{2}+\mathrm{4}+\mathrm{8}+\mathrm{16}+.....{n}\:{days}=\mathrm{2046} \\$$$$\frac{\mathrm{2}\left(\mathrm{2}^{{n}} −\mathrm{1}\right)\:}{\mathrm{2}−\mathrm{1}}=\mathrm{2046}\:\:\:\:\Rightarrow\mathrm{2}^{{n}} =\mathrm{1024} \\$$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{n}=\mathrm{10}\:{days} \\$$

Commented by mathocean1 last updated on 26/Jun/20

$${thank}\:{you}\:{sirs}... \\$$

Commented by mathocean1 last updated on 26/Jun/20

$${thank}\:{you}\:{sirs}! \\$$

Commented by smridha last updated on 26/Jun/20

$$\boldsymbol{{welcome}} \\$$

Answered by smridha last updated on 26/Jun/20

$$\mathrm{2}+\mathrm{2}^{\mathrm{2}} +\mathrm{2}^{\mathrm{3}} +\mathrm{2}^{\mathrm{4}} +....+\mathrm{2}^{\boldsymbol{{n}}} =\mathrm{2046} \\$$$$\frac{\mathrm{2}\left(\mathrm{2}^{\boldsymbol{{n}}} −\mathrm{1}\right)}{\left(\mathrm{2}−\mathrm{1}\right)}=\mathrm{2046}\:\Rightarrow\mathrm{2}^{\boldsymbol{{n}}} =\mathrm{1024}=\mathrm{2}^{\mathrm{10}} \\$$$$\boldsymbol{{so}}\:\boldsymbol{{n}}=\mathrm{10}\boldsymbol{{days}} \\$$