So I started working Polya's Mathematics and Plausible Reasoning, Volume 1: Induction and Analogy in Mathematics second question in chapter one. I highly recommend you purchase the book.

The problem states the following.

Consider the table:

Guess the general law suggest by these examples, express it in suitable mathematical notation, and prove it.

This is a lot harder than the previous question because there is an equation, so we have to look at both sides. If we look at the left side first we notice that each line is increased by 2 elements. First we have 1 element (namely the number 1), then we jump to 3 elements (1+ 2+ 3). then again to 7 elements (10 + 11 + 12 + 13 + 14 + 15 + 16).

If we analyze this fact alone we know that the progression is some form of

while

I tried some more analysis by taking the difference between sums, but that yielded no positive result. So I switched for the time being to the other side. On the right side we have a more recognizable pattern. Looking at 1, I started to think what could possible equal 1 other than itself. The first thing that came to mind is any number raised to the zero power.

,

this did not help me much but it was pointing me in the right direction, for exponents are the right way to go. I then looked at second line,

,

and thought about what base numbers could equal both of those numbers alone. I started with base 2 because its the smallest base I know that does not equal 1. If we look at the base two system we can see that

.

Now we are getting somewhere, I found two of my numbers and only looked at one base system (namely base 2).The next line,

,

gives me further clues to the pattern, we have a repeat of the highest power form the previous line and a new highest power added on. Now a base 2 system gets me the number 8 but it does not yield me 27, which is base 3

.

This leads me to believe that pattern may take fold in base 2 and base 3 numbers. 1,8,27,64 all can be found with

.

That last base 2 number,

,

kind of bothered me cause the exponent jumped so high. We went from 0,3,6. I was starting to think I only had to deal with base 2 and base 3, but no combination of exponents could net me the sum of the next line is the sequence from the left side of the equation.If we followed the pattern on the left, the next line should look like

,

with 9 elements that sum to 189. I can't make that number with base 2 and base 3 alone. So went back to the third line on the right side of the equation and pondered on the number 64. Then it came to me,

.

Now a pattern was being formed clearly for me too see. We are dealing with a cubic expression. The right hand side comes out be in the form of

,

if you test with each line of the problem it works out. The first line is

.

The next line is

.

The third line is

.

The final line comes out to be

.

The pattern checks out. Now that we finished that pattern up we still have to go back to the left side of the equations. This one was a bit tricky and I kept playing around with summations to try and get right combination of numbers. I used double summations, single summations, arithmetic sequences, and any other trick I could think of. I got absolutely no where with this part of the problem....$%&*#&@! I finally cracked and checked the back of the book for solutions, turns out the right side of the equation was correct. The bad news is I have no idea how he got the left hand of the equation.

So, I had to reverse engineer it. The book tells us

is the solution to the left hand side. I can somewhat understand the equation except for that last quadratic

.

Where in the hell did that come from, and why in the world was it hanging out in some kind of summation? So basically this little guy exiles me back to my white boards. I started back to practicing my summations, I started with the most basic summation

.

I worked at a few of those up to 6 elements, then I think about it. It occurs me that

need not have to be part of the summation. So I start to play with

.

I want some kind of constant to represent n with out changing, so I made k = n. I came up with

.

I run the expression through n = 0,1,2. It failed, miserably I might add. So I modified the equation to read

,

this worked for any

.

The only problem is I can't figure out how they got the first element, I have to assume its there and it works. Overall, the experience was good. Half of my intuition was correct and the other half wrong. I know the left hand side has to do with some arithmetic progression, but this is all I could come up with. I am satisfied that my skill has got me the basic pattern as usual, but I still need to work on producing the exact equations faster.

Reference:

- Polya, George.
*Mathematics and Plausible Reasoning Volume 1: Induction and Analogy in Mathematics*. Vol. 1. Princeton, New Jersey: Princeton University Press, 1954. 1-8. 2 vols. Print.