Can't solve a sequence (to determine if a given value is a member)

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Discussion Overview

The discussion revolves around determining whether the value 41/81 is a member of a specific mathematical sequence defined by the formula a_n = (n^2 + 1) / (2n^2). Participants explore methods to solve for n and analyze the properties of the sequence, including its monotonicity and limits.

Discussion Character

  • Mathematical reasoning
  • Exploratory
  • Homework-related

Main Points Raised

  • One participant asks for help in proving if 41/81 is part of the sequence defined by a_n = (n^2 + 1) / (2n^2).
  • Another participant suggests rearranging the equation to find n such that (1/2) + (1/2n^2) = 41/81.
  • Several participants discuss the method of cross-multiplying to solve the equation.
  • A participant expresses uncertainty about their calculations after cross-multiplying and distributing terms.
  • Another participant confirms the equation derived from cross-multiplying and encourages further steps to solve for n.
  • Participants discuss the implications of finding n, with one concluding that 41/81 corresponds to the 9th element of the sequence.
  • Questions arise about exploring the monotonicity of the sequence and how to graph it.
  • Some participants discuss the behavior of the sequence as n increases, suggesting it approaches a limit of 1/2.
  • One participant expresses feelings of inadequacy regarding their understanding of sequences and higher-level mathematics.

Areas of Agreement / Disagreement

Participants generally agree on the steps to solve for n and the conclusion that 41/81 is the 9th term of the sequence. However, there is ongoing discussion about the monotonicity and limits of the sequence, with no consensus on how to formally prove these properties.

Contextual Notes

Limitations include the participants' varying levels of understanding of mathematical concepts, which affects their ability to engage with the problem fully. Some steps in the mathematical reasoning remain unresolved, particularly regarding the formal proof of the sequence's behavior as n approaches infinity.

Who May Find This Useful

This discussion may be useful for students or individuals interested in mathematical sequences, problem-solving techniques, and the exploration of limits and monotonicity in sequences.

  • #31
MarkFL said:
I would think the argument I gave in my post previous to this would suffice then.

I agree...

But,now i have new problem,about Mathematical induction,should i open new Thread?
 
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  • #32
andreask said:
I agree...

But,now i have new problem,about Mathematical induction,should i open new Thread?

Yes, and I would post it in our Discrete Mathematics sub-forum.
 
  • #33
MarkFL said:
Yes, and I would post it in our Discrete Mathematics sub-forum.

I will...

Maybe you can take a look...if you want...
 
  • #34
I still have one question...What kind of type this sequence is? Arithmetic or geometric? I can't figure it out...
 
  • #35
andreask said:
I still have one question...What kind of type this sequence is? Arithmetic or geometric? I can't figure it out...

An arithmetic sequence has the property:

$$a_{n+1}-a_{n}=d$$

While a geometric sequence would have the property:

$$\frac{a_{n+1}}{a_{n}}=r$$

Note: Both $d$ and $r$ are constants that do not depend on $n$. Does this sequence satisfy either property?
 
  • #36
MarkFL said:
An arithmetic sequence has the property:

$$a_{n+1}-a_{n}=d$$

While a geometric sequence would have the property:

$$\frac{a_{n+1}}{a_{n}}=r$$

Note: Both $d$ and $r$ are constants that do not depend on $n$. Does this sequence satisfy either property?

It seems its arthmetic
 
  • #37
andreask said:
It seems its arthmetic

I find:

$$a_{n+1}-a_{n}=\frac{(n+1)^2+1}{2(n+1)^2}-\frac{n^2+1}{2n^2}=-\frac{2n+1}{2n^2(n+1)^2}$$

Thus, the difference between two succeeding terms is a function of $n$, and so the sequence is not arithmetic.
 
  • #38
MarkFL said:
I find:

$$a_{n+1}-a_{n}=\frac{(n+1)^2+1}{2(n+1)^2}-\frac{n^2+1}{2n^2}=-\frac{2n+1}{2n^2(n+1)^2}$$

Thus, the difference between two succeeding terms is a function of $n$, and so the sequence is not arithmetic.

Ad its not geometric?
 
  • #39
andreask said:
And its not geometric?

What do you find when you compute the ratio I gave above?
 
  • #40
MarkFL said:
What do you find when you compute the ratio I gave above?

Will try tomorrow,im litlle busy now
 

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