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Concept explainers
Show that if and arc integers with , then, Conclude that the Pascal triangle is symmetric with respect to a vertical line drawn from the topmost entry.
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To prove: To show that and are integers with , then .
Answer to Problem 46AYU
The last really simple thing to observe is that Pascal’s triangle is symmetric about the central axis. If we draw a vertical line down the page from the 1 in the 0 row, what appears on the left side of the triangle is the same as what appears on the right hand side of the triangle. This is because of the relation .
Explanation of Solution
Pascal’s triangle is a sequence of rows that form a triangle that gets wider as we go down. Let’s label the rows as 0, 1, 2, . . . etc. For example:
Here 0 row is formed by taking the element . The 1 row is listing the elements and . The 2 row is listing the elements , and and so on.
The n row becomes . Further position the elements and so that the element is between them and one level above them.
As it stands, it is quite cumbersome to compute the rows of Pascal’s triangle. For example, to compute the 8 row, we have to compute each element from . Instead of doing such hard work, let’s look at the table with the numbers computed up through the 8 row and look for patterns
Note that there are a few easy things to observe.
First, the entries on the left and right edges are alwalys 1. This is easy to explain, because they are always of the form or and
Further, the entries next to the edges in each row just give the number of the row. For example, in the 6 row, the entries next to the 1’s on the outer edge are 6 and 6. This is because they represent and . More generally, in the row, these entries are and , so this is why we always get for those entries.
The last really simple thing to observe is that Pascal’s triangle is symmetric about the central axis. If we draw a vertical line down the page from the 1 in the 0 row, what appears on the left side of the triangle is the same as what appears on the right hand side of the triangle. This is because of the relation .
Chapter 12 Solutions
Precalculus Enhanced with Graphing Utilities
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