Can Geometric Progressions Starting from One Sum to a Perfect Square?

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Discussion Overview

The discussion revolves around the question of whether it is possible to create a series of whole numbers in geometric progression starting from one that sums to a perfect square. Participants explore examples, propose methods, and share insights related to this mathematical inquiry.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant provides an example of a geometric progression: 1 + 3 + 9 + 27 + 81 = 121, which equals 11^2, suggesting that such progressions can sum to a perfect square.
  • Another participant mentions the difficulty in finding real-life examples of geometric progressions that sum to a perfect square, reiterating the initial inquiry.
  • A suggestion is made to consider the sequence of squares and the differences between subsequent squares as a potential method to find solutions.
  • One participant recommends using a spreadsheet to explore more examples, noting that there are two three-digit examples available.
  • A link to a Wikipedia page on geometric progressions is shared, presumably to provide formulas and further information relevant to the discussion.

Areas of Agreement / Disagreement

Participants express varying levels of success in finding examples, with some providing specific instances while others indicate challenges. The discussion remains unresolved regarding the generality of the claim and the existence of more examples.

Contextual Notes

Some participants' claims depend on specific examples and may not generalize. There are unresolved questions about the conditions under which geometric progressions can sum to perfect squares.

Medtner
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"Write out a series of three or more different whole numbers in geometric progression, starting from one, so that the numbers should add up to a square. So like, 1 + 2 + 4 + 8 + 16 + 32 = 63 (one short of a square)"(can't find an actual real life example)

I can't seem to find an answer for this?
 
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Just use a spreadsheet to try some more, there are two three-digit examples.
 
1+3+9+27+81=121=11^2
 
Medtner said:
"Write out a series of three or more different whole numbers in geometric progression, starting from one, so that the numbers should add up to a square. So like, 1 + 2 + 4 + 8 + 16 + 32 = 63 (one short of a square)"(can't find an actual real life example)

I can't seem to find an answer for this?
Try considering the sequence of squares, 0, 1, 4, 9, ..., and take the difference between subsequent squares. That will help you find the answer.
 

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