9 = ( 10 - 1) /9 =(10¹ - 1) / 9
99 = ( 100 - 1) / 9 =( 10² - 1 ) / 9
999 = ( 1000 - 1) / 9 =(10³ - 1 ) / 9
..................
9999 ...999 ( cu 2014 cifre) = ( 10²⁰¹⁴ - 1) / 9
99999..999 ( cu 2015 cifre ) = ( 10²⁰¹⁵ - 1 ) / 9
9999...999 ( cu 2014 cifre ) + .... + 999 + 99 + 9 =
= [ 10²⁰¹⁴ - 1 + 10²⁰¹³ - 1 + ... + 10³ - 1 + 10² -1 + 10¹ - 1] /9=
- 1 de 2014 ori
= [10²⁰¹⁴ +... 10³ + 10² +10 - 2014 ] / 9
= 10 ·[10²⁰¹⁴ - 1 ] / ( 10 -1) · 9 =
= [ 10¹⁺ ²⁰¹⁴ - 10 ] / 81 = [ 10²⁰¹⁵ - 10 ] / 81
b = ( 10²⁰¹⁵ - 1) / 9 - [ 10²⁰¹⁵ - 10 ] / 81 =
=[ 9 · ( 10²⁰¹⁵ - 1 ) - 10²⁰¹⁵ + 10 ] / 81=
= [ 9 ·10²⁰¹⁵ - 9 - 10²⁰¹⁵ + 10 ] / 81 =
= [ 8· 10²⁰¹⁵ + 1 ] / 81 = [ 8 ·10000...00000 + 1 ] / 81
= 800000..001 / 81 = 9 ·88888......9 / 81 =
= 88888888....889 / 9 = 888888....889 / 3²
8 de 2014 ori
dar 88888...8889 = nu este pp
