An imaginary infinite geometric tree

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Question from Elise, a teacher:

An imaginary infinite geometric tree grows 1m the first day.
2nd day 2 branches and right angles to each other and each 0.5 m long.
3rd day two new branches at ends of each of previous days' 2 branches, again at right angles, and only .25m long each.
And so on, infinitely.
Q: Use relationships of right-angled triangles and high school level knowledge of geometric series to show the tree height is limited to (4 + sqrt2)/3 m and width to (2(sqrt2 + 1))/3 m.

Elise,

You will need the additional assumption that the pairs of branches are symmetrically placed (so making an angle of 135 degrees) with respect to the branch they grow from.

Using this, draw a picture, and sum geometric series separately for the vertical and diagonal edges.

Good Hunting!
RD

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