Question

20 DE PUNCTE fie numarul n=11...1+22..2+.......+99....9 fiecare numar de forma AA...A continand cate 2015 cifre A .Determinati cate cifre de 9 contine numarul n

Answer 1

[tex]n=111..1+222..2+333...3+...+99...9, \underbrace{AA...A}\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~de\ 2015\ ori\\ n=11..1+2\cdot11..1+3\cdot 11..1+....+9\cdot 11..1\\ n=11..1(1+2+....+9)\\ n=\frac{10^{2016}-1}{9}\cdot\frac{9\cdot 10}{2}\\ n=5\cdot(10^{2016}-1)\\ n=5\cdot \underbrace{99...99}\\ ~~~~~~~~~de\ 2015\ ori\\ n=49999....95,unde\ cifra\ 9\ apare\ de\ 2014\ ori.\\ R:2014 ori [/tex]