# Finding the General Statement

• Jun 11th 2011, 08:21 AM
MScofield
Finding the General Statement
Consider the five rows of numbers....
1 1
1 3/2 1
1 6/4 6/4 1
1 10/7 10/6 10/7 1
1 15/11 15/9 15/9 15/11 1

Let En(r) be the (r+1)th element in the nth row, starting with r=0. Example: E5(2) = 15/9

Find the General Statement for En(r).
Discuss the scope and limitations of the general statement.
I found the general statement by looking at the nth term of the numerators as this was dependent upon only n. I then found the difference between the num and dem in terms of n and r, so I could express it as a single term. It came to...
En(r) = (n^2+n)/(n^2+n-2rn+2r^2)
I believe this is correct. Nevertheless, I also need to find this general statement graphically and am uncertain of how to accomplish this. I would appreciate any suggestions.

I understand a limitation is that the general statement is valid only for
r<(or equal to) n. However, I have heard there are more limitations and would appreciate any suggestions.

Thank you in advance. (This is a more refined version of a question I previously posted, which received no responses. I have tried to make it more appealing this time.)
• Jun 11th 2011, 09:22 AM
Opalg
Quote:

Originally Posted by MScofield
Consider the five rows of numbers....
1 1
1 3/2 1
1 6/4 6/4 1
1 10/7 10/6 10/7 1
1 15/11 15/9 15/9 15/11 1

Let En(r) be the (r+1)th element in the nth row, starting with r=0. Example: E5(2) = 15/9

Find the General Statement for En(r).
Discuss the scope and limitations of the general statement.
I found the general statement by looking at the nth term of the numerators as this was dependent upon only n. I then found the difference between the num and dem in terms of n and r, so I could express it as a single term. It came to...
En(r) = (n^2+n)/(n^2+n-2rn+2r^2)
I believe this is correct. Nevertheless, I also need to find this general statement graphically and am uncertain of how to accomplish this. I would appreciate any suggestions.

I understand a limitation is that the general statement is valid only for
r<(or equal to) n. However, I have heard there are more limitations and would appreciate any suggestions.

Thank you in advance. (This is a more refined version of a question I previously posted, which received no responses. I have tried to make it more appealing this time.)

I assume that 16/4 is a typo for 6/4, as indicated above.

You seem to be approaching this problem in the right way, but not totally accurately. In row n, the numerator is the n'th triangular number $\tfrac12n(n-1)$. If you make a table of the differences between the numerators and the denominators then it looks like this:
0 0
0 1 0
0 2 2 0
0 3 4 3 0
0 4 6 6 4 0
0 5 8 9 8 5 0

(I have put zeros at the start and end of each row on the grounds that 1 = n/n, so if the numerator is n then the denominator should also be n.) If you look down the diagonals of that diagram, you see that first element after the initial 0 in row n is equal to n, then next element after that is 2(n–1), then 3(n–2) and so on. The (r+1)th element is r(n–r+1).

That tells you that $E_n(r) = \frac{\frac12n(n-1)}{\frac12n(n-1) - r(n-r+1)}.$

When you simplify that fraction, it looks similar to your suggested answer, but not quite the same.

There is another thing that you should do, namely to check that the formula still works when r=0 and r=n.
• Jun 11th 2011, 10:14 AM
MScofield
Okay, thank you very much, but do you have any suggestions as to how I could find that general statement graphically?
• Jun 11th 2011, 10:35 AM
Opalg
Quote:

Originally Posted by MScofield
Okay, thank you very much, but do you have any suggestions as to how I could find that general statement graphically?

I don't know what "graphically" means in that context. You could plot the numbers in row n on a graph and see that they look as though they are on some sort of smooth curve, but that isn't going to help you to find the formula for $E_n(r).$

Confession: Your formula for $E_n(r)$ was right and mine was wrong. Sorry about that. I was confusing row n with row n+1.
• Jun 11th 2011, 11:45 AM
MScofield
I could be wrong on this, but I'm assuming that in order to find the general statement graphically I need to plot the element number (r) against the denominator for each row (n). I can then derive the equation of the graph from its vertex (through completion of the square). If I draw graphs for each row then I should observe that
y=r^2-nr+numerator.

By amalgamating (is there a mathematical term for this) the numerator formula and this formula for y I should end up with (n^2+n)/(n^2+n-2rn+2r^2), as above.

Is this a correct method? If so, then what are its scope and limitations?
Thanks again.