Difference between revisions of "1986 AIME Problems/Problem 7"
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Revision as of 18:04, 4 July 2013
Problem
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.
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. 
is equal to 
, so in binary form we get 
. However, we must change it back to base 10 for the answer, which is 
.
Solution 2
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 
See also
| 1986 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
- AIME Problems and Solutions
- American Invitational Mathematics Examination
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
