1986 AIME Problems/Problem 7

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Problem

The increasing sequence $1,3,4,9,10,12,13\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $100^{\mbox{th}}$ term of this sequence.

Solution

Solution 1

Rewrite all of the terms in base 3. Since the numbers are sums of distinct powers of 3, in base 3 each number is a sequence of 1s and 0s (if there is a 2, then it is no longer the sum of distinct powers of 3). Therefore, we can recast this into base 2 (binary) in order to determine the 100th number. $100$ is equal to $64 + 32 + 4$ , so in binary form we get $1100100$ . However, we must change it back to base 3 for the answer, which is $3^6 + 3^5 + 3^2 = 729 + 243 + 9 = 981$ .

Solution 2

Notice that the first term of the sequence is $1$ , the second is $9$ , the fourth is $27$ , and so on. Thus the $64th$ term of the sequence is $729$ . Now out of $64$ terms which are of the form $729$ + $'''S'''$ , $32$ of them include $243$ and $32$ do not. The smallest term that includes $243$ , i.e. $972$ , is greater than the largest term which does not, or $854$ . So the $95$ th term will be $972$ , then $973$ , then $976$ , then $977$ , and finally $981$

1986 AIME (Problems • Resources)
Preceded by Problem 6	Followed by Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15