# Thread: Pattern Problem

1. ## Pattern Problem

I just covered arithmetic sequences and how to develop algebraic rule for the pattern to prepare my 7th graders for their state test. The next question they faced was similar, but it wasn't an arithmetic sequence.

The pattern was:
1
5
14
30
55
....etc

So you are adding:

2^2
then 3^2
then 4^2

There was a nice picture of a pyramid of cans so you could see the new cans represented by 2 squared, and 3 squared. I assume because there is a pattern, there has to be an algebraic rule to finding the number of cans of the nth term but am unsure what it is.

I know the new cans are n squared and you add them to the total number of cans from the previous term, but am not sure how to represent that in an equation.

Is it really complicated, am I just not seeing it? Help!

2. ## Re: Pattern Problem

Originally Posted by Jman115
I just covered arithmetic sequences and how to develop algebraic rule for the pattern to prepare my 7th graders for their state test. The next question they faced was similar, but it wasn't an arithmetic sequence.
The pattern was:
1
5
14
30
55
....etc
I have never been close to a middle school classroom.
So this answer maybe useless tor 7th graders.

But this sequence is $\displaystyle a_N=1^2+2^2+3^2+\cdots+N^2$.

3. ## Re: Pattern Problem

Hello, Jman115!

I just covered arithmetic sequences and how to develop algebraic rules
for the pattern to prepare my 7th graders for their state test.
The next question they faced was similar, but it wasn't an arithmetic sequence.

The pattern was: .$\displaystyle 1,\:5,\:14,\:30,\:55\:\hdots$

So you are adding: .$\displaystyle 2^2,\text{ then }3^2\text{, then }4^2,\:\hdots$

There was a nice picture of a pyramid of cans,
so you could see the new cans represented by $\displaystyle 2^2,\:3^2\:\hdots$
I assume because there is a pattern.
There has to be an algebraic rule to finding the number of cans
of the $\displaystyle n^{th}$ term but am unsure what it is.

I know the new cans are $\displaystyle n^2$, and you add them
to the total number of cans from the previous term,
but am not sure how to represent that in an equation.

Is it really complicated, am I just not seeing it? Help!

Plato gave you the form of the sum of the first $\displaystyle n$ squares.

There is a formula, but it is probably above seventh-grade ability.

. . . . . $\displaystyle S_n \:=\:\frac{n(n+1)(2n+1)}{6}$

4. ## Re: Pattern Problem

Plato:

I was wondering if that was it, but I was expecting something simpler I guess. It seems like it would be a pain to find the 100th term using that formula because you would have to square so many numbers and not just plug the number 100 in like you can with an arithmetic sequence.

Soroban:

Having never seen that formula could you explain to my why it works. I can compute it but have no idea where it came from. I won't be able to even begin to explain it to them unless I can undertand where it came from. Some of my kids may be able to understand it if I can grasp it. Some of them are crazy smart. Others not so much.

Thanks for all the help guys, I love this place.

6. ## Re: Pattern Problem

Originally Posted by Jman115
Soroban:
Having never seen that formula could you explain to my why it works. I can compute it but have no idea where it came from. I won't be able to even begin to explain it to them unless I can undertand where it came from. Some of my kids may be able to understand it if I can grasp it. Some of them are crazy smart.
You must understand that these are not middle-school problems.
Here are two well-known theorems:
$\displaystyle \sum\limits_{k = 1}^N k = \frac{{N\left( {N + 1} \right)}}{2}$
and $\displaystyle \sum\limits_{k = 1}^N {k^2 } = \frac{{N\left( {N + 1} \right)\left({2N + 1} \right)}}{6}$.

7. ## Re: Pattern Problem

Originally Posted by Plato
You must understand that these are not middle-school problems.
Here are two well-known theorems:
$\displaystyle \sum\limits_{k = 1}^N k = \frac{{N\left( {N + 1} \right)}}{2}$
and $\displaystyle \sum\limits_{k = 1}^N {k^2 } = \frac{{N\left( {N + 1} \right)\left({2N + 1} \right)}}{6}$.

What I am trying to understand is why you multiply the nth term, by the nth term plus 1, by twice the nth term plus 1 then divide by six.

Where did all that come from, how did someone just figure that? Seems odd to me when thinking about it for what it is.

I know I have seen and worked with that formula before but it has been so long my mind is very fuzzy.

8. ## Re: Pattern Problem

Well, go here; lotsa reading:
http://www.macalester.edu/~bressoud/pub/CBN2.pdf

Google "sum squares formula" for more!

9. ## Re: Pattern Problem

I understand the first image. It is the second image I want to understand. I know the second is
related to the first but I am not sure how.