What is the general term of this sequence?

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    General Sequence Term
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Discussion Overview

The discussion revolves around identifying the general term of a specific sequence: 1, 5/3, 1, 15/17, 1, 37/35, 1, 63/65, etc. Participants explore potential patterns in the numerators and denominators, considering both mathematical expressions and relationships to perfect squares.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant requests assistance in finding the general term of the sequence.
  • Another participant notes that there are infinitely many general expressions for sequences that begin similarly and suggests that the numerator follows a pattern of alternating values, while the denominator lacks a clear pattern.
  • A different participant proposes considering perfect squares as a potential avenue for understanding the sequence.
  • One participant expresses confusion about how perfect squares relate to the sequence.
  • Another participant provides examples of how specific terms in the sequence can be expressed in relation to perfect squares, indicating a possible pattern in the numerators and denominators.
  • A participant acknowledges a shift in focus from powers of 2 to perfect squares, recognizing their relevance to the denominators.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the general term of the sequence, and multiple competing views regarding the patterns in the numerators and denominators remain present.

Contextual Notes

Participants express uncertainty about the patterns in the denominators and the overall structure of the sequence, indicating that further exploration is needed to clarify these aspects.

Denisse
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Could you help me to find the general term of the sequence:

## 1 , \frac{5}{3} , 1 , \frac{15}{17} , 1 , \frac{37}{35} , 1 , \frac{63}{65} ,... ##

Thank you!
 
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There are infinitely many general expressions for series which begin like that.
If you look for the "easiest" expression, there could be something simple, but I don't see it at the moment.
The numerator seems to follow the pattern (same as denominator, 2 more, same, 2 less, same, 2 more, ...).
I don't see a clear pattern for the denominator, however.
 
mfb said:
I don't see a clear pattern for the denominator, however.

Think about perfect squares.
 
micromass said:
Think about perfect squares.

Perfect squares? How? I don't see it
 
Well, 5/3 is (2^2 + 1)/(2^2 - 1) and 15/17 = (4^2 - 1)/(4^2+1) and 37/35 = ( 6^2 +1)/(6^2-1) and so on and so forth.
 
micromass said:
Think about perfect squares.
Oh, nice. I was too focused on powers of 2, which are close (+1, or +3 in one case) to all of the "visible" denominators.
 

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