Find the 100th term of a given sequence.

anemone

MHB POTW Director
Staff member
The increasing sequence 1, 3, 4, 9, 10, 12, 13 ... consists of all those positive integers which are powers of 3 or sum of distinct powers of 3. Find the 100th term of this sequence.

caffeinemachine

Well-known member
MHB Math Scholar
The increasing sequence 1, 3, 4, 9, 10, 12, 13 ... consists of all those positive integers which are powers of 3 or sum of distinct powers of 3. Find the 100th term of this sequence.
Let $n=(a_ka_{k-1}\ldots a_1a_0)_2$. Then the $n$-th number in the sequence is $3^ka_k+3^{k-1}a_{k-1}\cdots+3^1a_1+a_0$. It hinges on the fact that $3^{r+1}>1+3+\cdots+3^r$. Therefore the 100th number is $3^6+3^5+3^2$.

anemone

MHB POTW Director
Staff member
Let $n=(a_ka_{k-1}\ldots a_1a_0)_2$. Then the $n$-th number in the sequence is $3^ka_k+3^{k-1}a_{k-1}\cdots+3^1a_1+a_0$. It hinges on the fact that $3^{r+1}>1+3+\cdots+3^r$. Therefore the 100th number is $3^6+3^5+3^2$.
Hi caffeinemachine, thanks for participating in the problem and your answer is correct. I want also to say that at MHB, we are never short of talented maths experts here, and I've gained some very useful insights from the site, and for this I am so thankful! caffeinemachine

Well-known member
MHB Math Scholar
Hi caffeinemachine, thanks for participating in the problem and your answer is correct. I want also to say that at MHB, we are never short of talented maths experts here, and I've gained some very useful insights from the site, and for this I am so thankful! Keep such problems coming anemone!

anemone

MHB POTW Director
Staff member
Keep such problems coming anemone!
Hey caffeinemachine, are you one of the ardent fans of sequences and series? caffeinemachine

Well-known member
MHB Math Scholar
Hey caffeinemachine, are you one of the ardent fans of sequences and series? Not really.. In the olympiad genre I am an 'ardent fan' of Combinatorics and Geometry questions. In curriculum mathematics I like Discrete math the most, esp Algebra and Combinatorics. And among the mathematical disciplines in which I am a newbie I find Relativity and Non Euclidean Geometry the most fascinating and Romantic.

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Hey caffeinemachine, are you one of the ardent fans of sequences and series? anemone

MHB POTW Director
Staff member
Not really.. In the olympiad genre I am an 'ardent fan' of Combinatorics and Geometry questions. In curriculum mathematics I like Discrete math the most, esp Algebra and Combinatorics. And among the mathematical disciplines in which I am a newbie I find Relativity and Non Euclidean Geometry the most fascinating and Romantic.

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For me, my interests lie in trigonometry and sequences and series, which I think is a shameful thing to just like mathematics in these two narrow fields of mathematics... 