How Is the Sequence 2, 0, 2/27, 0, 2/125 Formulated?

  • Context: Undergrad 
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Discussion Overview

The discussion revolves around the formulation of the sequence 2, 0, 2/27, 0, 2/125, exploring how the zeros are incorporated and the underlying pattern of the non-zero terms. Participants also draw parallels with another sequence, 1, 4, 1, 16, 1, 36, discussing similar methods of defining terms based on their positions.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • One participant suggests that the sequence can be expressed as 2/n^3 for the odd indexed terms, questioning how the zeros are derived.
  • Another participant proposes that every other term being zero could be represented by the expression 1 - (-1)^n in place of the 2.
  • Further discussion introduces a second sequence, 1, 4, 1, 16, 1, 36, where a similar pattern is noted, with odd indexed terms being defined as 1 and even indexed terms as 4n^2.
  • Participants discuss defining the function f(n) based on whether n is odd or even, with one stating that for odd n, f(n) = 1, and for even n, f(n) = 4(m)^2, where m is an integer.
  • There is a suggestion that the expression for even indexed terms could also be simplified to n^2.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the exact formulation of the sequences or the best way to express the zeros. Multiple approaches and definitions are proposed, indicating ongoing exploration and debate.

Contextual Notes

The discussion includes various assumptions about the sequences and their definitions, with participants not fully resolving the mathematical steps or the implications of their proposed formulations.

Alexx1
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Can someone help me with this sequence?

2 0 2/27 0 2/125

If I only look at the odd numbers it's: 2/n^3

But I don't know how they get that zero..
 
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Could be just every other term is 0.

A convenient way to write that would be 1 - (-1)^n in place of the 2.
 


CRGreathouse said:
Could be just every other term is 0.

A convenient way to write that would be 1 - (-1)^n in place of the 2.

Thank you very much!
 


CRGreathouse said:
Could be just every other term is 0.

A convenient way to write that would be 1 - (-1)^n in place of the 2.

Can you help me with this sequence also?

1 4 1 16 1 36

If I only look at the odd numbers it's: n^2

But I don't know how they get to '1'..
 


Alexx1 said:
Can you help me with this sequence also?

1 4 1 16 1 36

If I only look at the odd numbers it's: n^2

But I don't know how they get to '1'..

Again, just define it that way: f(2n + 1) = 1, f(2n) = 4n^2. You can use the same trick with (-1)^n if you want -- make the exponent 0 for odds and 1 for evens.
 


CRGreathouse said:
Again, just define it that way: f(2n + 1) = 1, f(2n) = 4n^2. You can use the same trick with (-1)^n if you want -- make the exponent 0 for odds and 1 for evens.

How do you define f(n)?
 


He just did. If n is odd, n= 2m+ 1 for some integer m so f(n)= 1, if n is even, n= 2m for some integer m so f(n)= f(2m)= 4(m)^2. Of course, if n= 2m then m= n/2 so you could also write this as 4(n/2)^2= 4n^2/4= n^2.
 

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