Counting with powers of three
Number Theory
Level
3
All positive integers which can be expressed as a sum of one or more different integer powers of 3 are written in increasing order. The first six terms are 1, 3, 4. 9, 10 and 12. What is the 100th term?
The answer is 981.
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1 solution
Every number that is a sum of 1 or more distinct powers of 3 has only 1s and 0s in it's base 3 representation. the 100th positive integer we can make with only 1s and 0s is 100 in binary, or 1100100. Interpreting 1100100 in base 3 instead, we find our answer of 981.