# 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.

- - - Updated - - -