Discover the rule and prove it

[diredragon]

Homework Statement

Discover what the rule is and prove it. (Picture attached)

Homework Equations

Everything is allowed

The Attempt at a Solution

This is a problem from a little book of fun math problems i own. I discovered three rules.
Given the number of a row you want to study ( marked N ) the following rules tell you:
N(N-2)+2 = which number starts the process
2N-1 = how many elements does it have
(N-1)^3+N^3 = sum of all numbers in that row.
My question is: how am i going to prove this? Do i need to prove all three rules?

 

[Dr. Courtney]

Can you be more specific than "everything is allowed" regarding relevant equations?

 

[diredragon]

Can you be more specific than "everything is allowed" regarding relevant equations?

I mean any equation is allowed as the goal is to prove the rule you think is appropriate to describe the given system. I don't know if i got the rule correct but it seems to me that i did. I found three rules connected by the row number. Maybre there is a different rule...i can't see it,...

 

[Ray Vickson]

Homework Statement

Discover what the rule is and prove it. (Picture attached)

Homework Equations

Everything is allowed

The Attempt at a Solution

This is a problem from a little book of fun math problems i own. I discovered three rules.
Given the number of a row you want to study ( marked N ) the following rules tell you:
N(N-2)+2 = which number starts the process
2N-1 = how many elements does it have
(N-1)^3+N^3 = sum of all numbers in that row.
My question is: how am i going to prove this? Do i need to prove all three rules?

When you say "discover what the rule is", that leaves the issue: rule for what?

Some of the "rules" could refer to the things you have discovered already, but that leaves out some others. For example, you could ask for a formula to find the number $n$ that lies in row $r$ and position $p$ of that row. Or, you could ask for a formula to find the number $n$ that lies in row $r$ and position $p$ of that row.

 

[diredragon]

It just says find the rule and prove it probably referring to the sum of the numbers in a row, bit the other rules would be fun to find aswell. How would i prove the sum rule and is it correct?

 

[haruspex]

It just says find the rule and prove it probably referring to the sum of the numbers in a row, bit the other rules would be fun to find aswell. How would i prove the sum rule and is it correct?

As you say, there are three rules to be discovered, but there is only one that can be proved.
You have to discover (or rather, guess) what rules are used to generate the two columns of numbers. But that is not intended to be difficult. They might just as well have given you those rules, expressed in words. It was just easier to express them by example.
You first task is to turn each of those rules into an algebraic expression. Writing that the two expressions are equal constitutes discovering the third rule. Finally, you are to prove that those two expressions are equal.

 

[diredragon]

So are the three rules ones i posted in the first post and only the sum rule can be proven?

 

[haruspex]

So are the three rules ones i posted in the first post and only the sum rule can be proven?

Yes. You cannot prove that a given finite sequence of numbers will continue to follow any particular pattern. You can only check that the part of the sequence given satisfies it.

 

[RUber]

So, the pattern in the triangle will be assumed to be true.
Given the pattern, you want to show:
$\sum_{\text{row } n} = \sum_{i= N(N-2)+2}^{N^2} i = N^3 + (N-1)^3$
For any N.
Alternatively, you could say:
$\sum_{i= 1}^{(N+1)^2} i - \sum_{i= 1}^{(N)^2} i = (N+1)^3 + (N)^3$

 

[diredragon]

So, the pattern in the triangle will be assumed to be true.
Given the pattern, you want to show:
$\sum_{\text{row } n} = \sum_{i= N(N-2)+2}^{N^2} i = N^3 + (N-1)^3$
For any N.
Alternatively, you could say:
$\sum_{i= 1}^{(N+1)^2} i - \sum_{i= 1}^{(N)^2} i = (N+1)^3 + (N)^3$

So its:
(N^2+1) + (N^2+2) + ... + (N+1)^2 = N^3 + (N+1)^3
Row with just number 1 is the 0th row and the others follow. N stands for the row number. How to prove this?

 

[haruspex]

So its:
(N^2+1) + (N^2+2) + ... + (N+1)^2 = N^3 + (N+1)^3
Row with just number 1 is the 0th row and the others follow. N stands for the row number. How to prove this?

Yes. What methods have you been taught for establishing such relationships?

 

[RUber]

I would recommend showing it directly using the alternate phrasing in post #9.
Do you know, or can you find a rule for, $\sum_{n=1}^K n = ?$
Using that, it should just be a matter of expanding and cancelling to show that they match up.

 

[diredragon]

Yes. What methods have you been taught for establishing such relationships?

I have learned the proof by induction. Can that be used here? If not can you recommend a methos for me to learn so to come back and prove this.

I would recommend showing it directly using the alternate phrasing in post #9.
Do you know, or can you find a rule for, $\sum_{n=1}^K n = ?$
Using that, it should just be a matter of expanding and cancelling to show that they match up.

I don't know what you mean by this. What rule are you referring to? What is K in the post?

 

[SammyS]

I have learned the proof by induction. Can that be used here? If not can you recommend a method for me to learn so to come back and prove this.
Im don't know what you mean by this. What rule are you referring to? What is K in the post?

It means the sum of the integers from 1 through K . There's a formula for that.

 

[haruspex]

I have learned the proof by induction. Can that be used here? If not can you recommend a methos for me to learn so to come back and prove this.
I don't know what you mean by this. What rule are you referring to? What is K in the post?

Yes, induction can be used. As RUber and Sammy write, the formula for the sum of consecutive integers is one you probably should know. If you know it, you can use that instead of induction. If you don't know it, or don't want to assume it without proof, proving it would be via induction anyway, so it still comes down to induction in the end.

 

[RUber]

From: https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss
Anecdotes[edit]
There are several stories of his early genius. According to one, his gifts became very apparent at the age of three when he corrected, mentally and without fault in his calculations, an error his father had made on paper while calculating finances.

Another story has it that in primary school after the young Gauss misbehaved, his teacher, J.G. Büttner, gave him a task: add a list of integers in arithmetic progression; as the story is most often told, these were the numbers from 1 to 100. The young Gauss reputedly produced the correct answer within seconds, to the astonishment of his teacher and his assistant Martin Bartels.

Gauss's presumed method was to realize that pairwise addition of terms from opposite ends of the list yielded identical intermediate sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on, for a total sum of 50 × 101 = 5050. However, the details of the story are at best uncertain (see[41] for discussion of the original Wolfgang Sartorius von Waltershausen source and the changes in other versions); some authors, such as Joseph Rotman in his book A first course in Abstract Algebra, question whether it ever happened.

 

[diredragon]

Yes, i know the formula and i came up with something but i don't know if its exactly a proof.
The sum of consecutive integers is (n(n+1))/2 where n stands for number of integeres in a row. Number of integeres in a row can be found in any sequence row from above using the formula
((N+1)^4+(N+1)^2)/2 - (N^2(N^2+1))/2 = N^3 + (N+1)^3 (from post 9)
Simplifying gets the both sides to match. Is this proof enough or is necessary to prove also that the left side truly shows the number of integers in a given row.

 

[RUber]

You can formalize it by way of induction.
It works for row 1, assume it is true for row N, and demonstrate that it must be true for row N+1.

 

[diredragon]

((N+2)^2*((N+2)^2+1))/2 - ((N+1)^2*((N+1)^2+1))/2 = (N+1)^3 + (N+2)^3 it works for case N+1

 

[RUber]

So then you know the rule, and you know the method. Now, you just need to write up the proof so that it is clear to the reader.
a) state how the pyramid is constructed ... the n'th row ends in n^2.
b) state your claim regarding the right side ( sum of the n'th row is n^3 - (n-1)^3 )
c) demonstrate the rule holds for n = 1.
d) assume the rule holds for an arbitrary row n, and show that it must be true for row n+1.
In lieu of induction, you could directly state that the sum of the nth row, which starts at (n-1)^2+1 and goes to n^2 , is equal to the right side for any n.