Question

calculati 9+99+999+9999+....+999999.....9999(2012 cifre)
HEEELP

Answer 1

9+99+999+....99....99 9 (2012 cifre)=
9+1+99+1+999+1+....99..99(2012 cifre)+1-2012=
10+100+1000+100...000 (2012de 0)-2012=111...1110 (cu 2012de1)-2012=
111.....11109098 incare am 2012-4=2008de 1 la inceput

un numar fara nici un sens

Answer 2

[tex]9+99+999+9999+....+999999.....9999 $ \{2012 cifre\}$ = \\ = 10-1 + 100-1 + 1000-1 + 10000-1 + ...+ 1000....000 $\{2012 de 0$)}-1= \\ =10+100+1000+10000+....+1000..000 $\{2012 de 0\}$ - 1\cdot 2012= \\ =10^1+10^2+10^3+10^4+...+10^{2013} - 2012 \\ \\ $Notam S = 10^1+10^2+10^3+10^4+...+10^{2013} $ \Big|\cdot 10\\ \\ 10\cdot S = 10^2+10^3+10^4+10^5+....+10^{2014} \\ 10\cdot S = 10^2+10^3+10^4+10^5+....+10^{2013}+10^{2014} + 10^1 - 10^1 \\ 10\cdot S = 10^1+10^2+10^3+10^4+10^5+....+10^{2013}+10^{2014} - 10^1 [/tex]

$10\cdot S = S+10^{2014} - 10 \Rightarrow 10\cdot S - S = 10^{2014}-10 \\ \\ 9\cdot S = 10^{2014}-10 \Rightarrow S = \dfrac{10\cdot(10^{2013}-1)}{9} \\ \\ \\ \Rightarrow9+99+999+9999+....+999999.....9999 (2012 cifre) = S - 2012 = \\ \\ = \dfrac{10\cdot(10^{2013}-1)}{9} - 2012$