MathDB

Consider the sequence

(a_n)_{n\ge 1}

defined by

a_n=n

for

n\in \{1,2,3.4,5,6\}

, and for

n \ge 7

a_n={\lfloor}\frac{a_1+a_2+...+a_{n-1}}{2}{\rfloor}

where

{\lfloor}x{\rfloor}

is the greatest integer less than or equal to

x

. For example :

{\lfloor}2.4{\rfloor} = 2, {\lfloor}3{\rfloor} = 3

and

{\lfloor}\pi {\rfloor}= 3

.
For all integers

n \ge 2

, let

S_n = \{a_1,a_1,...,a_n\}- \{r_n\}

where

r_n

is the remainder when

a_1 + a_2 + ... + a_n

is divided by

3

. The minus

-

denotes the ''remove it if it is there'' notation. For example :

S_4 = {2,3,4}

because

r_4= 1

1

is removed from

\{1,2,3,4\}

. However

S_5= \{1,2,3,4,5\}

betawe

r_5 = 0

and

0

is not in the set

\{1,2,3,4,5\}

. 1. Determine

S_7,S_8,S_9

and

S_{10}

. 2. We say that a set

S_n

for

n\ge 6

is well-balanced if it can be partitioned into three pairwise disjoint subsets with equal sum. For example :

S_6 = \{1,2,3,4,5,6\} =\{1,6\}\cup \{2,5\}\cup \{3,4\}

and

1 +6 = 2 + 5 = 3 + 4

. Prove that

S_7,S_8,S_9

and

S_{10}

are well-balanced . 3. Is the set

S_{2019}

well-balanced? Justify your answer.

4