I figured for Memorial Day I would contribute a small portion of mathematical knowledge. I cracked open Polya's Mathematics and Plausible Reasoning, Volume 1: Induction and Analogy in Mathematics. Its been a long time since my last post from his chapter 1, so I thought it was time to add some more problems to the list. The third problem states:

- Observe the values of the successive sums
- Is there a simple rule?

The first thing I would do is start noting all observations of whats given:

- All sums are of odd natural numbers, we must be careful of which definition of 'natural numbers' we use. I am using the one thought to me in the United States, so we don't include 0 as a natural number.

- Next I would write down what I think is next in the sequence, we can safely guess the next set of numbers because we are just adding the next consecutive odd number to the summation:

- I would also write down the actual sums of these summations, so I would evaluate each one and record the answer:

From here we obtain a new list of numbers that give us another clue of whats going on. Working on the summations, I use the fact that we are only summing odd numbers. If you remember the formula for an odd number is:

The summation is of odd numbers, which would look like this:

Any given 'n' from 1 - 8 would produce any of the listed sums previously stated. Now we can focus on the sequence of evaluated sums:

You may not recognize the pattern right away, it was not until I saw the numbers 25, 36, 49, and 64 that I recognized the pattern. These four numbers in consecutive order signify to me that we are dealing with 'squares'. 4 and 9 are not so easy to point out because they are in the beginning of sequence. Now I know that our formula

Looks like this

The last thing we should do is verify what we found, is what I just explained correct? Looking in the back of the book, the author gives this solution:

I was right, the only difference is I wrote the left hand side in sigma notation. This problem seemed a little easier than his previous problem, but I still learned something by working the problem. Now we can move onto the fourth problem, the **fourth problem** states:

- Observe the values of the successive sums
- Is there a simple rule?

Working the last problem, this problem may seem even easier because we already went through the steps for a similar problem. I will literally go through the exact same steps, I try to be consistent in my work. Naturally you might feel this is a waste of time to do the same steps on paper, you might be inclined to just skip ahead and find the answer. I would advise against that notion of thinking. Repetition is a fine way of honing your mathematical skills, I have seen a lot of math students feel like this idea is beneath them. I even seen a lot of high level math students question my constant need to work out each problem, they just stood there perplexed as I worked out such trivial steps in problems.

For me, the hunt for understanding a problem at its core is the highest reward. So while some mathematicians can excel at solving complicated problems that I can not, I can understand basic algebra where they tend to look confused. This is a personal observation, it is in no means a judgment on the reader.

Back to our problem, we start at step one. The first thing I would do is start noting all observations of whats given:

- All the sums include both odd and even numbers, again 0 is not included. So I can safely assume we are only working with natural numbers. Remember, I am using the definition of 'natural numbers' that does not include 0.

- Next I would write down what I think is next in the sequence, this step would require that I know the pattern already. It is not as easy as just adding consecutive odd numbers. I noticed by experience that numbers are in the form of cubes.

- , , , ,

- Now we can guess the next set of summations:

- I would also write down the actual sums of these summations, so I would evaluate each one and record the answer:

From here we obtain a new list of numbers that give us a clue of whats going on. This is a bit different from the last problem, all of sudden I don't think its going to be so trivial for us. Looking at the sequence of evaluated sums, I started to notice that each element is square of some number. What tipped me off was the 36 and 100, I noticed that these are just 6 and 10 raised to the 2nd power respectively. To test this theory I took the square roots of 225, 441, and 784. Each element yielded me a perfect square.

, , , , , ,

From this observation I noted a new sequence of numbers, I noted the numbers that are raised to the 2nd power:

I had seen this exact sequence of numbers some where before, but I could not put my finger on it. I started to take the difference between each element, trying to find some sort of pattern.

This technique did not yield me much fruit, I had to think of other ways to find a pattern. One way to find patterns with number theory that helps me is to represent the numbers visually.

This technique did not yield me much fruit, I had to think of other ways to find a pattern. One way to find patterns with number theory that helps me is to represent the numbers visually.

- I had the most amazing thing happen after this post was posted. Somebody found a mistake, thanks to the reader Brian. When I first read his reply on how there must have been a mistake, I was confused. He pointed out a point of view I had not thought of before. The idea he pointed out was that when you take the difference of each element in the sequence you get in order the natural numbers. I will demonstrate: our sequence looks like this 1,3,6,10,15,21,28.

- , , , , , ,

- You may wonder why I subtracted 0 from 1, even though 0 is not in the sequence of numbers we are currently looking at. The simple answer is there is no element before the 1, so I had no number to subtract from 1. The only way I could represent that is by subtracting 0, it also helps us put the number 1 in our new sequence. If you look at the results of these differences, you will see the natural counting numbers 1-7 in order. So we get another new sequence of numbers:

- This is what Brian was talking about when he said I must be mistaken, which made me think. Why was Brain saying I must have been mistaken? I don't think the pattern is clear to see. Later on, I started to think about why I had disagreed with Brian. I don't just disagree with people for the sake of disagreeing with people, there had to be a reason for my concern. It later came to me that we were talking about two different things, the reason for this was my poor skill in communicating my thoughts.

- When I said taking the difference between the elements yielded no fruit, I need another way to find a pattern. I miss spoke, I should not have written that. There was a pattern to the technique, as Brian pointed out earlier. I saw the pattern in my work, then I just put it aside. The problem for me, at the time, was I did not know the relevance of the pattern to the triangle numbers.

- My experience with triangle numbers had been mostly with computation, because I was looking for perfect squares at the same time (about a year ago). Thanks to Brian's response I can finally say I know a pattern that just links to triangle numbers.

- Before, when I worked on triangle numbers I only knew the sequence 1, 3, 6, 10, 15, 21, 28. I also knew that if you visually represent these numbers in terms of dots you can make triangles. Each row of the triangle can only hold its index of dots. So the first row can only have one dot, the second row can only have two dots, the third row can only have three dots, the fourth row can only hold 4 dots. You can see a picture here.

- If you draw all these triangles on a piece of paper (only a few them), then count the number of dots per triangle you will get the above mentioned sequence. So the first triangle has one dot, the second triangle has three dots, the third triangle has 6 dots.

- This is where the big explosion of understanding came to my mind, I finally understood the relevance in the difference of elements for the sequence:

- If you look at the difference of the elements, you will see that you just get the natural counting numbers in order. This observation is what Brian was talking about from the very beginning. You will also notice visually that each triangle in succession has exactly one extra row compared to the previous triangle. This extra row on the bottom is the exact difference between each triangle. So visually you can see the sequence

- The bottom row of the second triangle in the picture that you see has exactly two dots, the third triangle's bottom row has exactly 3 dots. So now you can see the difference visually and arithmetically.

- My problem was that I had lacked the proper skill, I had found away to increase my pattern recognition skill. When I was searching for patterns, I was looking for differences similar to arithmetics sequences. For example in the sequence:

- The difference is always 2, so you can conclude that the pattern is increasing by 2. I could have used an example where the numbers increased by 3, or decreased by 5. All these examples have a constant difference, and the natural counting numbers has a difference of just 1. The pattern just reminded me of an index sequence, I had not given it much observation. So when I found the pattern to our problem, I just put it aside because I did not know how to use it just at that moment. I instead relied on my previous experiences to help me out. I put in a lot of effort to solving these types of problems, so I have an easier time of recognizing these problems when they recur again. This is how I ended up writing about finding the answer below:

I started to write each number in terms of "dots". You can do this at least for the elements that are less than 20. Then it hits me, this is the exact kind of work I did in my number theory book by Silverman. The question in that book deals with perfect squares and triangle numbers.

To verify my guess, I put the sequence "1, 3, 6, 10, 15, 21, 28" into the google search engine and it pointed me to triangle numbers. Lucky for us, the Wiki page has explicit formula:

So now we can finally answer the question properly, the left hand side is nothing more than summation of cubes. The right hand side is the triangle number formula raised to the 2nd power. So our answer looks like this:

I can't check my answer in the back of the book on this one, because the author just points to some other page further in the book. I will look to my mathematical confidence to assure me that this answer is sufficient enough. These two problems are great examples of working out a problem and seeing the major differences, even though they look similar to each other. I am glad I worked out both problems in the same fashion, it allowed me to see the subtle differences that helped me solve the problem.

References:

- Polya, George.
*Mathematics And Plausible Reasoning: Induction And Analogy In Mathematics*. 6th ed. Vol. 1. Princeton, New Jersey: Princeton University Press, 1954. 1-9. 2 vols. Print. - Melchoir, . "First six triangular numbers.svg."
*Wikipedia, the free encyclopedia*. N.p., 9 Oct. 2011. Web. 28 May 2013. <http://en.wikipedia.org>. Path: http://en.wikipedia.org/wiki/File:First_six_triangular_numbers.svg. *Wikipedia, the free encyclopedia*. N.p., n.d. Web. 28 May 2013. <http://en.wikipedia.org>. Path: http://en.wikipedia.org/wiki/Square_number.