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### Finding the nth Term

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Date: 03/30/2002 at 07:40:43
From: Kathy Angus
Subject: Finding the nth term

This is the investigation. This was the starting point:

A colony of bees is making a honeycomb consisting of hexagonal cells.
round the center cell, and on the third day they add the remaining
ring of cells in the picture. Every day they add a new ring of cells
right round the honeycomb.

The investigation posed two questions:
1. How many cells will there be at the end of the sixth day?
2. On which day will the number of cells pass the 1000 mark?

I managed to find the formula for this.
I extended this by using 3D shapes instead of 2D shapes. I used cubes,
with day 1 starting with 1 cube; on day 2 I surrounded the first cube
with more cubes, both the sides and edges, completely surrounding it.
I.e., making a 3x3x3 cube on day 2.

I continued this for each day: 3x3x3, 5x5x5, 7x7x7, etc..
I wanted to find the relation between the number of days and the
number of NEW ADDITIONAL cubes. These are the results I got:

Number of days(n)	New additional days(y)
1                         0
2                        26
3                        98
4                       218
5                       386

The formula I  got was: Y = (2n-1)3-(2n-3)3

This works for all except the first day! I would be grateful for any

Thanks, Kathy
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Date: 03/30/2002 at 22:51:05
From: Doctor Peterson
Subject: Re: Finding the nth term

Hi, Kathy.

I would have said that on day 1 you have added 1 block, rather than
zero, since you probably had none before beginning.

Further, since on each day the total is a complete cube, as on the
first and second days, the total on day N is (2N-1)^3, so the number
added on day N would be

(2N-1)^3 - (2(N-1)-1)^3

which is just what you got. (I see now that you did not get this
formula from the numbers, but directly from the problem, which is
good!) This can be simplified to

[8N^3 - 12N^2 + 6N - 1] - [8N^3 - 36N^2 + 54N - 27] =

24N^2 - 48N + 26

I would check this by making a table including the total each day:

day     total     new   formula
0     0^3=  0      -       -
1     1^3=  1      1       2
2     3^3= 27     26      26
3     5^3=125     98      98
4     7^3=343    218     218

Now, we can see why day 1 would be anomalous: on the other days, we
started with a cube, and added two to each dimension, going to the
next odd cube. But on the first day, we didn't start with a cube -1
unit on a side; we started with nothing, and added a single cube. That
is, the number on the first day did not arise by following the same
rule of surrounding an existing cube, so it is not surprising that

Notice that my formula for the _total_ applies to day 1, but not to my
fictional day 0; so that your formula for the number of blocks added
applies only _after_ day 1, since it depends on the previous day's
total.

It is not unusual in mathematics to require a starting condition apart
from the formula that applies elsewhere. You just have to say that the
formula holds for N > 1. In some cases it makes some sort of sense to
continue the pattern back in time before you started, but here that
doesn't work, so we don't need to be troubled by the formula not being
extensible to N < 2.

Incidentally, I don't think your description of the process by which
you are building larger blocks is quite right. You are not just adding
to the faces and edges, but also to the corners. If you added only
cubes that touch on a face, you would have a considerably more
interesting problem, since the shapes you are building would not be
mere cubes.

But the question you raised turns out to be very interesting, and
deserves the thought we've given it! It's all too easy to ignore
starting conditions and imagine that all formulas work for all cases,
and checking against reality is a very good practice to avoid this.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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Date: 04/01/2002 at 11:34:09
From: Kathy Angus
Subject: Finding the nth term

Hello,

I would like to thank you very much for the help you gave me with the
investigation. It was a great help. I will be using this great site
again in the future! Thanks for the quick response!

Thanks once again, Kathy
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Associated Topics:
College Discrete Math
High School Discrete Mathematics

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