MathDB

6

Part of 2009 Argentina National Olympiad

Problems(1)

a_{n+1}=m a_n where m is integer , S_n= the sum of digits of a_n

Source: 2009 Argentina OMA Finals L3 p6

1/15/2023
A sequence a0,a1,a2,...,an,...a_0,a_1,a_2,...,a_n,... is such that a0=1a_0=1 and, for each n≥0n\ge 0 , an+1=m⋅ana_{n+1}=m \cdot a_n , where mm is an integer between 22 and 99 inclusive. Also, every integer between 22 and 99 has even been used at least once to get an+1a_{n+1} from ana_n . Let SnSn the sum of the digits of ana_n , n=0,1,2,...n=0,1,2,... . Prove that Sn≥Sn+1S_n \ge S_{n+1} for infinite values ​​of nn.
sum of digitsDigitsrecurrence relationnumber theory