Track / Stage 5 / 32 of 400 #632 of 2000

Problem 632

AIME late

Number theory Difficulty 5.1 Find the answer

$8 \cdot 1$ Arrange all powers of 3 and the sums of any finite number of distinct powers of 3 in an increasing sequence:
$1,3,4,9,10,12,13, \cdots$

Find the 100th term of this sequence.

Your answer

Official solution

［Solution］Let the $n$ -th term of the sequence be denoted as $a_{n}$ . Clearly, $a_{n}$ is a positive integer, and its ternary representation uses only the digits 0 and 1. Conversely, any natural number whose ternary representation uses only the digits 0 and 1 must be a term in this sequence. Therefore, the binary representation of $n$ is exactly the ternary representation of $a_{n}$ . Since the binary representation of 100 is 1100100, we have
$a_{n}=3^{6}+3^{5}+3^{2}=981 .$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.