1986 AIME Problems/Problem 7
The increasing sequence consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the term of this sequence.
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. is equal to , so in binary form we get . However, we must change it back to base 3 for the answer, which is .
Notice that the first term of the sequence is , the second is , the fourth is , and so on. Thus the term of the sequence is . Now out of terms which are of the form + , of them include and do not. The smallest term that includes , i.e. , is greater than the largest term which does not, or . So the th term will be , then , then , then , and finally
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