SAT1000 - P876

Number Theory Level 2

Given that an infinite sequence $\{a_n\}$ consists of $k$ distinct values, $S_n=\displaystyle \sum_{i=1}^n a_i$ .

If $\forall n \in \mathbb N^+$ , $S_n \in \{2,3\}$ , then find the maximum of $k$ .

Have a look at my problem set: SAT 1000 problems

The answer is 4.

1 solution

Alexander Shannon
Jun 21, 2020

$a_1$ is either $2$ or $3$ , so we can have $S_1\in \{2,3\}$ . In either case, if $S_n=2$ , the options for $a_{n+1}$ are $\{0,1\}$ , and if $S_n=3$ , the options for $a_{n+1}$ are $\{0,-1\}$ . Therefore, we can have only $4$ distinct values $\{0,-1,+1,a_1\}$ , where $a_1=2$ or $a_1=3$ .

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Alexander Shannon - 11 months, 3 weeks ago

SAT1000 - P876

The answer is 4.

1 solution

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