Project B06 - Pascal's Triangle

1. Refer to the attached sheet labeled Sheet 1.
   1. Start with the hexagon marked with the number 1 and determine the number of downward paths to the center of each hexagon. Consider only those paths that pass through the centers of hexagons. An example of a downward path is shown.
   2. It is possible to determine these numbers without counting, but by recognizing a pattern. Carefully describe this pattern.
2. Determine a function S(n) that gives the sum of the numbers across row n.
   1. Compute the sum of the numbers across the 20th row.
   2. Show that 2S(n) = S(n+1).
   3. For which row do the numbers add up to 16384?
3. Copy the numbers from Sheet 1 to Sheet 2.
   1. For each hexagon on Sheet 2, place a ruler along each edge that rises from left to right. Draw each line that passes through one of the leftmost hexagons. These lines are usually called ``shallow diagonals.'' Two shallow diagonals are drawn for you.
   2. Determine the sum of the numbers along each shallow diagonal. This sequence of numbers is called the Fibonacci sequence (or the sequence of Fibonacci numbers).
   3. Describe the relationship between terms in the Fibonacci sequence.
   4. Describe a situation where the Fibonacci sequence arises in nature.
4. Find the sequence of numbers 1,3,6,10,15,21,.... These numbers are called triangular numbers. Why?
5. The triangular array of numbers that you've been studying is called Pascal's triangle. Write a short biography of this Pascal person.
6. Research Pascal's triangle. Describe another pattern found in this triangle.
7. Describe a problem or application in which Pascal's triangle naturally arises.