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