Pascal's Triangle and the Fibonacci Sequence
by Owen Borville
November 16, 2021
Science, Mathematics, Nature
A triangular-shaped display of whole numbers beginning at the top with 1 and adding in value downward where:
Each number below is the sum of the two numbers directly above it diagonally:
Is known as Pascal's Triangle, named after the French mathematician Blaise Pascal but was also previously studied by other mathematicians in Asia and Europe.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
The numbers displayed in Pascal's Triangle on each row are also the coefficients of the expansion of any binomial expression used in the binomial theorem.
The Fibonacci Sequence and Pascal's Triangle: a series of numbers representing the diagonal row sums of Pascal's Triangle also forms the Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34...
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
The sum of the numbers on each horizontal row are powers of 2: 1, 2, 4, 8, 16, ....
The numbers on the second diagonal row form counting numbers: 1, 2, 3, 4, 5, 6, 7, ...
The third diagonal row features triangular numbers...:1, 3, 6, 10, 15, 21, ...
The fourth diagonal row features tetrahedral numbers...: 1, 4, 10, 20, 35, ...
The Fibonacci Sequence is found in many places in nature in living things and non-living things, providing a link between mathematics and nature and showcasing the evidence for Intelligent Design in Nature.
by Owen Borville
November 16, 2021
Science, Mathematics, Nature
A triangular-shaped display of whole numbers beginning at the top with 1 and adding in value downward where:
Each number below is the sum of the two numbers directly above it diagonally:
Is known as Pascal's Triangle, named after the French mathematician Blaise Pascal but was also previously studied by other mathematicians in Asia and Europe.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
The numbers displayed in Pascal's Triangle on each row are also the coefficients of the expansion of any binomial expression used in the binomial theorem.
The Fibonacci Sequence and Pascal's Triangle: a series of numbers representing the diagonal row sums of Pascal's Triangle also forms the Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34...
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
The sum of the numbers on each horizontal row are powers of 2: 1, 2, 4, 8, 16, ....
The numbers on the second diagonal row form counting numbers: 1, 2, 3, 4, 5, 6, 7, ...
The third diagonal row features triangular numbers...:1, 3, 6, 10, 15, 21, ...
The fourth diagonal row features tetrahedral numbers...: 1, 4, 10, 20, 35, ...
The Fibonacci Sequence is found in many places in nature in living things and non-living things, providing a link between mathematics and nature and showcasing the evidence for Intelligent Design in Nature.
