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MATH 035
Penn State University
Dr. James Sellers
Handout: Pascal’s Triangle
We now shift gears to look at a set of topics related to work which was popularized by the
French mathematician and philosopher Blaise Pascal (1623-1662). In terms of mathematical
work, Pascal is probably best known for his work in elementary probability. As part of that
work, he spent a good deal of time working with what we now call Pascal’s Triangle. In fact,
Pascal wrote a book entitled Treatise on the Arithmetical Triangle which was the most
influential work on the topic.
As an aside, it should be noted that Pascal actually was NOT the first person to study the topic
(which I will describe to you in just a moment). But here is proof that Pascal’s Triangle was not
first studied by Pascal. (This image is dated in the early 14th century.)
OK, so what’s up with this Pascal’s Triangle? Well, let’s build it together.
Patterns in Pascal’s Triangle
Greatest Common Factor and Divisibility Properties
1. Find the greatest common divisor of each of the rows in the triangle (rows 2 through 12).
Of course, ignore the ones on the ends of the rows. What pattern(s) (if any) do you see?
2. What do these patterns imply about the divisibility of all the numbers in certain rows of
Pascal’s triangle?
3. Using a clean copy of the triangle, shade in all the values that are divisible by 3. What
geometrical shapes seem to appear?
4. Using a clean copy of the triangle, shade in all the values that are divisible by 5. What
geometrical shapes seem to appear?
5. Would you expect the same sort of shapes to appear if you performed #4 above with any
prime (including 2)?
Patterns in the Sums
We now want to consider some other patterns that arise very naturally in Pascal’s Triangle.
Here we focus on special sums in the triangle.
1. Determine a pattern that arises by alternately adding and subtracting all the numbers in
each row of Pascal’s triangle. Write a formula that describes this pattern. (This formula
should not be too complicated!)
2. Determine a pattern that arises by summing all the numbers in each row of Pascal’s
triangle. Write a formula that describes this pattern.
3. Pick one of the rows of the triangle. Perform the following sum: The first number in the
row plus two times the second number in the row plus four times the third number in the
row plus eight times the next number in the row and so on (until you have used all the
numbers in the row). Determine a pattern that arises by performing such summations.
Write a formula that describes this pattern.
4. How can you generalize the pattern you see in #2 and #3 above? (To start, perform exercise
#3 again but multiply by powers of 3 rather than by powers of 2.)
5. Gear change: Perform several sums along the “main” diagonals of the triangle. (For
example, consider 1 + 4 + 10 + 20 + 35.) Do you see a pattern?
6. One last sum pattern: Consider the very specialized sums which are built using the “gentler
sloping upward diagonals” (for lack of a better phrase). These would be the following sums:
1+2
1+3+1
1+4+3
1+5+6+1
What well-known sequence emerges?
© 2010, James A. Sellers