Question

Calculati suma: 9+99+999+...+ 99.... 9 . (de 1998 ori 9)

Answer 1

[tex]S=9+99+999+...+999...999\\ S=9(1+11+.....+11...11)\\ S=9(\frac{10-1}{10-1}+\frac{10^2-1}{10-1}+...+\frac{10^{1999}-1}{10-1})\\ [/tex]
[tex]S=9*\frac{1}{10-1}(10-1+10^2-1+....+10^{1999}-1)\\ S=\not9*\frac{1}{\not9}(\frac{10^{2000}-10}{9}-1999)\\ S=\frac{10^{2000}-18001}{9}[/tex]

Answer 2

9+99+999+...+99...9 = 10-1 + 100-1 +1000-1 + 10...0-1 (sunt 1998 de 0)=
= 10+100+1000+...+10....0 -(1+1+1+...+1) =
111...10 - 1998 =      (11...1 este de 1998 ori)
= 11...1109112  aici cifra 1 este de 1994 ori