Find # Sequences for Increasing Sequence Problem

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In summary: Yeah that's how the example worked, the new sequence must be non decreasing as well.Actually I looked at this again and for the easiest one, ie. the average sequence 1, 2, 3, 4, ... , there's 2 possible sequences that could have came from:0, 2, 2, 4, 4, 6, 6, ...or1, 1, 3, 3, 5, 5, ...
  • #1
trollcast
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Homework Statement


For an increasing sequence of numbers, how many other sequences could this be the average sequence of.

Homework Equations



Where the average sequence, a = 0.5( s + s[i+1] )

The Attempt at a Solution



If there's n terms in the original sequence.
The number of differences between consecutive terms is (n - 1)
Find all these, (n-1) and find the lowest difference.
Then this lowest difference + 1 is your answer?

eg.
s = 1, 3, 6, 10, 12
s[i+1] - s = 2, 3, 4, 2

The lowest difference here is 2 so there's 2 possible sequences for which s is the average of?
 
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  • #2
trollcast said:

Homework Statement


For an increasing sequence of numbers, how many other sequences could this be the average sequence of.

Homework Equations



Where the average sequence, a = 0.5( s + s[i+1] )

The Attempt at a Solution



If there's n terms in the original sequence.
The number of differences between consecutive terms is (n - 1)
Find all these, (n-1) and find the lowest difference.
Then this lowest difference + 1 is your answer?

eg.
s = 1, 3, 6, 10, 12
s[i+1] - s = 2, 3, 4, 2

The lowest difference here is 2 so there's 2 possible sequences for which s is the average of?


Hey trollcast! :smile:

Suppose we start with the simplest possible sequence we can think of: 1,2,3,4,...

Now we'd look for another sequence that has the same averages.
To get the same average, if we increase a specific number by some amount ε > 0, the next number must be decreased by that same amount ε.
So we'd add an alternating sequence ##(-1)^i ε##.
Since the resulting sequence still has to be increasing, that ε must be less than (1/2).

So we'd start with the sequence specified by ##s_i = i## and we'd end up with the sequence specified by ##s_i' = i + (-1)^i ε## with ##0 < ε < \frac 1 2 ##.

How many of those ε's are there?
 
  • #3
I like Serena said:
Hey trollcast! :smile:

Suppose we start with the simplest possible sequence we can think of: 1,2,3,4,...

Now we'd look for another sequence that has the same averages.
To get the same average, if we increase a specific number by some amount ε > 0, the next number must be decreased by that same amount ε.
So we'd add an alternating sequence ##(-1)^i ε##.
Since the resulting sequence still has to be increasing, that ε must be less than (1/2).

So we'd start with the sequence specified by ##s_i = i## and we'd end up with the sequence specified by ##s_i' = i + (-1)^i ε## with ##0 < ε < \frac 1 2 ##.

How many of those ε's are there?

Would there be 0 if the sequence has to be integers?
 
  • #4
trollcast said:
Would there be 0 if the sequence has to be integers?

Yes if we're talking about the sequence 1,2,3,...

But for another sequence like 0,10,20,30,40,...
we can find 1,9,21,29,41,...
or 2,8,22,28,42,...
or ...
 
  • #5
I like Serena said:
for another sequence like 0,10,20,30,40,...
we can find 1,9,21,29,41,...
or 2,8,22,28,42,...
or ...
Maybe I'm misreading the OP. If a = 0,10,20,30,40,.. is the given sequence, I thought we were looking for other sequences s s.t. a = 0.5( s + s[i+1] ). 1,9,21,29,41,... does not appear to be a solution.
If we start with an arbitrary s[0] then s[i+1] = 2a - s would appear to generate the rest uniquely. Perhaps it is also required that s is increasing (or maybe non-decreasing)?
 
  • #6
haruspex said:
Maybe I'm misreading the OP. If a = 0,10,20,30,40,.. is the given sequence, I thought we were looking for other sequences s s.t. a = 0.5( s + s[i+1] ). 1,9,21,29,41,... does not appear to be a solution.
If we start with an arbitrary s[0] then s[i+1] = 2a - s would appear to generate the rest uniquely. Perhaps it is also required that s is increasing (or maybe non-decreasing)?


Yeah that's how the example worked, the new sequence must be non decreasing as well.
 
  • #7
Actually I looked at this again and for the easiest one, ie. the average sequence 1, 2, 3, 4, ... , there's 2 possible sequences that could have came from:

0, 2, 2, 4, 4, 6, 6, ...

or

1, 1, 3, 3, 5, 5, ...

I've sat and mucked around with various other sequences and found that something like:

1, 2, 8, 9 has no sequences for which it could be the average but I can't figure out a way to work this out without going through and working out the possible values for each average sequence?
 

1. What is the "Find # Sequences for Increasing Sequence Problem"?

The "Find # Sequences for Increasing Sequence Problem" is a mathematical problem that involves finding all possible increasing sequences of a given length in a set of numbers. The goal is to determine the total number of sequences and identify each sequence.

2. What is the significance of solving this problem?

Solving the "Find # Sequences for Increasing Sequence Problem" can help in various applications such as data analysis, pattern recognition, and sequence prediction. It can also help in understanding mathematical concepts and improving problem-solving skills.

3. How do you approach this problem?

The first step is to understand the problem and its requirements. Then, one can use a systematic approach to generate and identify all possible increasing sequences of the given length in the given set of numbers. This can be done using algorithms or by hand depending on the complexity of the problem.

4. What are some common challenges in solving this problem?

Some common challenges in solving the "Find # Sequences for Increasing Sequence Problem" include identifying the correct starting point, handling duplicate numbers, and considering all possible combinations. It is also essential to have a clear understanding of the problem and the set of numbers given.

5. How can this problem be applied in real-life situations?

The "Find # Sequences for Increasing Sequence Problem" can be applied in various real-life situations, such as analyzing financial data, predicting stock market trends, and identifying patterns in biological sequences. It can also be used in coding and programming to optimize algorithms and improve efficiency.

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