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Comparing sequences
Teddy and Mitty looked at the challenge of Ivan and Yana:
http://www.weblabs.org.uk/wlplone/Members/Sofia/my_reports/Report.2005-02-03.0823
The sequences to be compared are:
(1) 2, 4, 8, 16, ...
(2) 1, 1024, 59049, ...
(3) 1, 4, 27, 256,...
Teddy: The number 1024 is 210. So my guess is that the sequence (2) has a general term n10 for n=1, 2, 3, ....
To chek this let's compute 310. That's it - just 59049.
The sequence (1) has a general term 2n and it is an old friend of ours.
And the sequence (3) is a challenge given by Ken. It is in fact the sequence { nn } - I love it!
George: So which one of the sequences grows fastest ?
Teddy:
Let's take n=10
- The term A10 for the three sequences will be 210, 1010, and 1010, respectively. This means that the second and the third sequence are equal and they are greater than the first sequence (210).
- Let us consider the sequences (2) and (3). If n=11 we have for the corresponding terms 1110 and 1111. Therefore sequence 3 is greater than sequence 2 and grows fastest.
Jenny: So, last time you said that a sequence is greater than another one if the corresponding terms are greater. And according to your definition sequence (3) is "greater" than sequence (2). But does this mean that it grows faster? And what do we mean by "growing faster"?
George: In fact Ivan wanted to compare two concrete sequences by asking if it is possible for the terms of the second sequence to surpass the corresponding terms of the first one. A further question in this line would be if two sequences can pass each other more than once. As for the growth of a sequence and how to measure and compare it, is an interesting but different question.
Let us observe how Ivan (who is in anothefr group) is trying to compare the three sequences:
Ivan: Ken's sequence (3) is easy: 1, 4, 27, 256,... i.e.
11, 22, 33, 44, ...
Of course it grows faster than sequences (1) and (2) since both the base and the power grow. The robot for the Ken's sequence is:
I built it in two steps and I wonder why Ken said he had used more...
But I wanted to compare the three sequences and here are the robots for them in one box:
Jenny: How would you like to compare them?
Ivan: I could use the scale.
George: But have in mind that your robot will compare only the first terms of the nests of (1) and (2)
Ivan: OK, I will make a robot sucking the terms on top of the nests…
Ah, the scale changes its position immediately after the first terms of (1) and (2), therefore (2) becomes greater than (1)
George: Are you sure that this will be forever? Let us check their behavior in Excel.
Ivan: I will study the differences between the corresponding terms of every two sequences. It’s difficult to read them – they are in very special form in Excel. Besides when the power is bigger than 323 Excel is helpless (#N/A). Let me try with the ratio instead of the difference. How interesting! Sequences (1) and (2) pass each other twice!!! First 21/110 > 1 but 22/210 < 1 and second – 258/5810 <1 but 259/5910 > 1. (See the link to the Excel document below.)
No wonder it was not easy for me figure out which sequence is greater..
.Comment
Try this instead...
Posted by:
Ken
at
21-02-05
You can get a ToonTalk robot to produce a text pad with each number of a sequence on a different line. You need to drop a number on an erased text pad at some point.
It is interesting to see the shape of the numbers if you blur your ideas so you can't see the digits. For example,
1
10
100
1000
10000
is a triangle. What you can notice by looking at the shape of the sequence when displayed this way? Can you tell which sequence is growing faster?
P.S. When I said my robot took 6 or 7 steps I was counting every tiny step. So it picked up a 1, dropped it, picked up Maggie, copied a number, typed ^, dropped the number operation, copied the result, dropped the copy on a bird, and finally dropped Maggie.