Need help finding the general statement

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June 25th 2012, 07:17 AM
max17

Need help finding the general statement

Hi guys first time posting here, not sure if it belongs in this category...

But I need help with an internal assessment.

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

Describe how to find the numerator of the sixth row. Using technology, plot the relation between the row number,n, and the numerator in each row. Describe what you notice from your plot and write a general statement to represent this.

Find the sixth and seventh rows. Describe any patterns you used.

Let E_n(R) be the (r+t)^th element in the n^th row, starting with r=o.

Example: E_5(2)=15/9
Find the general statement for E_n(r)

Test the validity of the general statement by finding additional rows. Discuss the scope/and or limitations of the general statement.

How would one go about this?
June 25th 2012, 02:04 PM
Soroban

Re: Need help finding the general statement

Hello, max17!

Here is some of it . . .

Quote:

$\begin{array}{cccccccccccc} (1) &&&&& 1 && 1 \\ \\[-4mm] (2) &&&& 1 && \frac{3}{2} && 1 \\ \\[-4mm] (3) &&& 1 && \frac{6}{4} && \frac{6}{4} && 1 \\ \\[-4mm] (4) && 1 && \frac{10}{7} && \frac{10}{6} && \frac{10}{7} && 1 \\ \\[-4mm] (5) &1 && \frac{15}{11} && \frac{15}{9} && \frac{15}{9} && \frac{15}{11} && 1 \end{array}$

Describe how to find the numerator of the sixth row.
Using technology, plot the relation between the row number, n, and the numerator in each row.
Describe what you notice from your plot and write a general statement to represent this.]

The numerator of the $n^{th}$ row is the $n^{th}$ Triangular Number: . $T_n \:=\:\frac{n(n+1)}{2}$

Quote:

Find the sixth and seventh rows. . Describe any patterns you used.
Let E_n(R) be the (r+t)^th element in the n^th row, starting with r=o.
What is R? . . . What is r? . . . What is t?

I believe they are:

$\begin{array}{cccccccccccccccccc}(6) && 1 && \frac{21}{16} && \frac{21}{13} && \frac{21}{12} && \frac{21}{13} && \frac{21}{16} && 1 \\ \\[-4mm] (7) & 1 && \frac{28}{22} && \frac{28}{18} && \frac{28}{16} && \frac{28}{16} && \frac{28}{18} && \frac{28}{22} && 1 \end{array}$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Here is my reasoning . . .

Read down the "first" diagonal.

$\begin{array}{cccccccccccc} (1) &&&&& . && \frac{1}{1} \\ \\[-4mm] (2) &&&& . && \frac{3}{2} && . \\ \\[-4mm] (3) &&& . && \frac{6}{4} && . && . \\ \\[-4mm] (4) && . && \frac{10}{7} && . && . && . \\ \\[-4mm] (5) &. && \frac{15}{11} && . && . && . && . \end{array}$

The denominators are: $1, 2, 4, 7, 11$
This is a quadratic function: $f(n) \:=\:\tfrac{1}{2}(n^2 - n + 2)$
The next two denominators are: $16\text{ and }22.$

Read down the "second" diagonal.

$\begin{array}{cccccccccccc} (1) &&&&& . && . \\ \\[-4mm] (2) &&&& . && . && \frac{3}{3} \\ \\[-4mm] (3) &&& . && . && \frac{6}{4} && . \\ \\[-4mm] (4) && . && . && \frac{10}{6} && . && . \\ \\[-4mm] (5) &. && . && \frac{15}{9} && . && . && . \end{array}$

The denominators are: $3, 4, 6, 9$
This is a quadratic function: $f(n) \:=\:\tfrac{1}{2}(n^2-3n+8)$
The next two denominators are: $13\text{ and }18.$

Read down the "third" diagonal.

$\begin{array}{cccccccccccc} (1) &&&&& . && . \\ \\[-4mm] (2) &&&& . && . && . \\ \\[-4mm] (3) &&& . && . && . && \frac{6}{6} \\ \\[-4mm] (4) && . && . && . && \frac{10}{7} && . \\ \\[-4mm] (5) &. && . && . && \frac{15}{9} && . && . \end{array}$

The denominators are: $6, 7, 9$
This is a quadratic function: $f(n) \:=\:\tfrac{1}{2}(n^2 - 5n + 18)$
The next two denominators are: $12\text{ and }16.$
June 25th 2012, 10:14 PM
max17

Re: Need help finding the general statement

Thank you! Helped a lot! I have one more question though. I need to find the limitations of the general statement, how would i formulate this?

Thank you!
June 26th 2012, 03:22 AM
max17

Re: Need help finding the general statement

I need to find the general formula F_n(r) giving me info about the r th element in the n th row. What is the connection between the 3 functions?
June 26th 2012, 10:18 PM
gasadg

Re: Need help finding the general statement

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