Consider the five rows of numbers shown below.1 11 3/2 11 16/4 6/4 11 10/7 10/6 10/7 11 15/11 15/9 15/9 15/11 1Describe how to find the numerator of the sixth row.Using technology, plot the relation...

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Consider the five rows of numbers shown below.1 11 3/2 11 16/4 6/4 11 10/7 10/6 10/7 11 15/11 15/9 15/9 15/11 1Describe 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 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) .Test the validity of the general statement by finding additional rows.Discuss the scope and/or limitations of the general statement.Explain how you arrived at your general statement.
Answered Same DayDec 20, 2021

Solution

David answered on Dec 20 2021
3 Votes
Consider the five rows of numbers shown below.
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 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) .
Test the validity of the general statement by finding additional rows.
Discuss the scope and/or limitations of the general statement.
Explain how you a
ived at your general statement.
Solution:
We can find the numerator of the given pattern above by understanding the Pascal‟s triangle
which is given as:
The numerators in the above given pattern for different rows are the elements shown in „blue‟ i.e.
the 3
d
element in the Pascal‟s triangle which are 1, 3, 6, 10, 15, 21,…….. The Pascal‟s triangle
shows a pattern of (


) (n+1)Cr. For our case such that
For (1+1)C2 = 2C2 = 1
For (2+1)C2 = 3C2 = 3
For (3+1)C2 = 4C2 = 6
For (4+1)C2 = 5C2 = 10
For (5+1)C2 =...
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