Pascal’s Triangle

The Fibonacci Series is found in Pascal’s Triangle.

Pascal’s Triangle, developed by the French Mathematician Blaise Pascal, is formed by starting with an apex of 1.  Every number below in the triangle is the sum of the two numbers diagonally above it to the left and the right, with positions outside the triangle counting as zero.

The numbers on diagonals of the triangle add to the Fibonacci series, as shown below.

Fibonacci numbers found in Pascal's Triangle

Pascal’s triangle has many unusual properties and a variety of uses:

  • Horizontal rows add to powers of 2 (i.e., 1, 2, 4, 8, 16, etc.)

  • The horizontal rows represent powers of 11 (1, 11, 121, 1331, 14641) for the first 5 rows, in which the numbers have only a single digit.

  • Adding any two successive numbers in the diagonal 1-3-6-10-15-21-28… results in a perfect square (1, 4, 9, 16, etc.)

  • It can be used to find combinations in probability problems (if, for instance, you pick any two of five items, the number of possible combinations is 10, found by looking in the second place of the fifth row.  Do not count the 1’s.)

  • When the first number to the right of the 1 in any row is a prime number, all numbers in that row are divisible by that prime number

Comments

      • says

        Wonderful video. I love approaching art and degisn from a maths and scientific angle and this illustrates that way of working perfectly. Plus, I only just noticed the link to further explanations so it’s even more exciting.Great post.

  1. Monica says

    the exterior of the triangle is made up of 1’s and the rest of the numbers are each the sum of their neighbours from the row above them. 2=1+1, 4=3+1, 21=6+15, etc.

  2. Vikrant says

    Hi,
    Can you explain how Pascal’s triangle works for getting the 9th & 10th power of 11 and beyond?

    Thanks,
    Vikrant

    • Parviz says

      if you see each horizontal row as one number (1,11,121,1331 etc.) it will show the powers of 11 just carry on the triangle and you should be able to find whatever power of 11 your looking for

  3. Mark says

    You can represent the triangle as a square. Rows & columns represent the decimal expension of powers of 1/9 (= o.111111 ; 1/81 = 0,0123456 ; 1/729 = 0.00136.)

  4. N says

    Hi, just wondering what the general expression for Tn would be for the fibonacci numbers in pascal’s triangle? Thanks

  5. john says

    the 2nd statement is not at all true, The horizontal rows represent powers of 11 (1, 11, 121, 1331, 14641, 1621051!=.15101051, etc…)
    only works for the first 5 rows
    11^0=1
    11^1=11
    11^2=121
    11^3=1331
    11^4=14641
    11^5=161051 is different than 15101051

  6. George Frank says

    Finding your presentation and explanation of Pascal’s Triangle was very interesting and its analysis amusing.
    What is remarkable is to find how each number fits in perfect order inside the triangular matrix to produce all
    those amazing mathematical relationships. Thank you so much..!!!

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