MHB Find the 100th term of a given sequence.

[anemone]

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]

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]

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]

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]

Keep such problems coming anemone!

Hey caffeinemachine, are you one of the ardent fans of sequences and series? (Nerd)

 

[caffeinemachine]

Hey caffeinemachine, are you one of the ardent fans of sequences and series? (Nerd)

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? (Nerd)

How about you? Your math interests?

 

[anemone]

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.

- - - Updated - - -How about you? Your math interests?

I see...

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...(Blush)