Question Number 160123 by mr W last updated on 24/Nov/21
$${if}\:{we}\:{write}\:{all}\:{natural}\:{numbers} \\ $$$${together}\:{like}\:{this} \\ $$$$\mathrm{1234567891011121314151617}..., \\ $$$${what}\:{is}\:{the}\:\mathrm{1000}{th}\:{digit}? \\ $$
Answered by nikif99 last updated on 24/Nov/21
$$\mathrm{1}−\mathrm{9}:\:\mathrm{9}\:{numbers}×\mathrm{1}\:{digit}=\mathrm{9}\:{digits} \\ $$$$\mathrm{10}−\mathrm{99}:\:\mathrm{90}×\mathrm{2}=\mathrm{180} \\ $$$$\mathrm{100}−\mathrm{369}:\:\mathrm{270}\:{numbers}×\mathrm{3}\:{digits}=\mathrm{810} \\ $$$$\Rightarrow\mathrm{1}−\mathrm{369}:\:\mathrm{999}\:{digits} \\ $$$$\mathrm{1000}{th}\:{digit}:\:\mathrm{3} \\ $$
Commented by mr W last updated on 24/Nov/21
$${thanks}! \\ $$
Answered by puissant last updated on 24/Nov/21
$$\mathrm{1}−\mathrm{9}\::\:\mathrm{9}\:{numbers}\:×\mathrm{1}\:{digit}\:=\:\mathrm{9}\:{digits} \\ $$$$\mathrm{10}−\mathrm{99}\::\:\left(\mathrm{99}−\mathrm{9}\right)×\mathrm{2}\:{digits}\:=\:\mathrm{180}\:{digits} \\ $$$$\mathrm{100}−\mathrm{999}\::\:\left(\mathrm{999}−\mathrm{99}\right)×\mathrm{3}\:{digits}\:=\:\mathrm{2700}\:{digits} \\ $$$$\left(\mathrm{1000}−\mathrm{9}−\mathrm{180}\right)=\left(\mathrm{1000}−\mathrm{189}\right)=\mathrm{811} \\ $$$${E}\left(\frac{\mathrm{811}}{\mathrm{3}}\right)+\mathrm{100}\:=\:\mathrm{370}..\:\mathrm{1000}{th}\:{digit}\::\:\mathrm{3} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...........\mathscr{L}{e}\:{puissant}......... \\ $$
Commented by mr W last updated on 24/Nov/21
$${thanks}! \\ $$