Find the 100th term of a given sequence.

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Discussion Overview

The discussion revolves around finding the 100th term of a specific increasing sequence composed of positive integers that are either powers of 3 or sums of distinct powers of 3. Participants explore the mathematical formulation and reasoning behind identifying terms in this sequence.

Discussion Character

  • Mathematical reasoning
  • Exploratory
  • Technical explanation

Main Points Raised

  • One participant describes the sequence as consisting of positive integers that are powers of 3 or sums of distinct powers of 3.
  • Another participant presents a formula for the n-th number in the sequence, relating it to binary representation and powers of 3.
  • A specific calculation is proposed for the 100th term, expressed as a sum of powers of 3: $3^6 + 3^5 + 3^2$.
  • Participants express appreciation for contributions and insights from others in the discussion.
  • There is a light-hearted exchange about personal interests in various areas of mathematics, including combinatorics, geometry, discrete math, and sequences.

Areas of Agreement / Disagreement

While some participants agree on the correctness of the proposed method for finding the 100th term, the discussion does not reach a consensus on the overall approach or the implications of the findings. Personal interests in mathematics vary among participants, indicating a diversity of perspectives.

Contextual Notes

The discussion includes references to mathematical concepts such as binary representation and the properties of powers of 3, but does not resolve the broader implications or potential limitations of the proposed methods.

Who May Find This Useful

This discussion may be useful for individuals interested in sequences, series, and mathematical reasoning, particularly those exploring combinatorial and algebraic concepts.

anemone
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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.
 
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anemone said:
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 $3^6+3^5+3^2$.
 
caffeinemachine said:
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!;)
 
anemone said:
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!
 
caffeinemachine said:
Keep such problems coming anemone!

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

- - - Updated - - -

anemone said:
Hey caffeinemachine, are you one of the ardent fans of sequences and series? (Nerd)
How about you? Your math interests?
 
caffeinemachine said:
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)
 

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