Pascal's Triangle

Pascal’s Triangle Patterns

1) Addition of the Rows: One of the interesting properties of the triangle is that
the sum of its rows is equal to 2n

where n corresponds to the number of the row:

1 = 1 = 20

1 + 1 = 2 = 21

1 + 2 + 1 = 4 = 22

1 + 3 + 3 + 1 = 8 = 23

1 + 4 + 6 + 4 + 1 = 16 = 24

2) Prime Numbers in the Triangle: Another pattern visible in the triangle deals with prime numbers. If a row starts with a prime number or is a prime numbered row, all the numbers that are in that row (not counting the 1’s) are divisible by that prime. If we look at row 5 (1 5 10 10 51), we can see that 5 and 10 are divisible by 5. However, for a composite numbered row, such as row 8 (1 8 28 56 70 56 28 8 1), 28 and 70 are not divisible by 8.

3) Fibonacci Sequence in the Triangle: By adding the numbers in the diagonals of the Pascal triangle the Fibonacci sequence can be obtained as seen in the figure given below.

There are various ways to show the Fibonacci numbers on the Pascal triangle. R. Knott was able to find the Fibonacci appearing as sums of “rows” in the Pascal triangle. He moved all the rows over by one place and here the sums of the columns would represent the Fibonacci numbers.

Properties of Pascal’s Triangle

• • Each number is the sum of the two numbers above it.
  • The outside numbers are all 1.
  • The triangle is symmetric.
  • The first diagonal shows the counting numbers.
  • The sums of the rows give the powers of 2.
  • Each row gives the digits of the powers of 11.
  • Each entry is an appropriate “choose number.”
  • And those are the “binomial coefficients.”