Facem cateva artificii de calcul:
1+11+111+....+111.111(de 9 ori)=
(10-1)/9+(10²-1)/9+(10³-1)/9+.....+(10¹⁰-1)/9=
(10+10²+10³+...+10¹⁰-9)/9=
Rezolvam suma din paranteza separat:
S=10+10²+10³+....+10¹⁰ |*10
10S=10²+10³+....+10¹¹
10S-S=10¹¹-10
S=10¹¹-10/9
Revenim:
(10¹¹-10/9-9)/9=
10¹¹-91/81=
10.....0 (11 de zero)-91/81=
9....909/81=123.456.789 (inainte de 0 sunt 8 de 9)
