PATTERNS IN PASCAL’S TRIANGLE

Generally if you look at the Pascal’s tringle some basic facts encountered are:

1. The first diogonal has 1’s.
2. The second diagonal has Natural numbers.
3. The third diagonal has Triangular numbers.
4. The fourth diagonal has Tetrahedral numbers.
5. The fifth diagonal has Pentalope numbers.

Power of 2

The sum of the nth row in the Pascal’s triangle is 2ⁿ.

Hockey stick pattern

The “Hockey Stick” pattern in Pascal’s Triangle is a distinctive geometric arrangement of numbers that resembles the shape of a hockey stick. This pattern emerges when you draw a diagonal line starting from the top of the triangle and then extend a horizontal line to the right of that diagonal. The number at the far end of the horizontal line is equal to the sum of all the numbers along the diagonal. This pattern is a notable feature of Pascal’s Triangle and is often used in combinatorics and mathematics to calculate specific combinations and sums.

Fibonacci Number

If you sum the numbers in each diagonal (starting from the top), you’ll get a sequence of numbers known as the Fibonacci sequence. For example, the sum of the numbers in the first diagonal is 1, the sum in the second diagonal is 1, the sum in the third diagonal is 2, and so on.

Symmetry

The triangle itself also possesses symmetry, resembling a mirror image, with numbers on the left side perfectly matching their counterparts on the right side.

Exponents of 11

The rows of Pascal’s Triangle represent the result of raising 11 to a certain power (exponent).

For example:

1. The first row of Pascal’s Triangle represents 11⁰, which equals 1.
2. The second row represents 11¹, which equals 11.
3. The third row represents 11², which equals 121.
4. The fourth row represents 11³, which equals 1331, and so on.

So, each row in Pascal’s Triangle corresponds to the result of raising 11 to a specific power.

From the 5th row, the answer will not be straight little calculation is needed to get the answer.

In the process of computation, commence your evaluation from the second digit from the left-hand side, extend it through to the fourth digit from the right-hand side, and ensure that the final three digits remain unaltered at the conclusion of the calculation.

The 5th row in Pascal’s triangle is 1 5 10 10 5 1 = 1 (5+1) (0+1)051=161051

The 6th row in Pascal’s triangle is 1 6 15 20 15 61 = 1(6+1)(5+2)(0+1)561 = 1771561

The 7th row from Pascal’s triangle is 1 7 21 35 35 21 7 1 = 1(7+2)(1+3)(5+3)(5+2)171 = 19487171.

and so on.

SIERPINSKI TRIANGLE

When applying a color scheme to distinguish odd and even numbers within Pascal’s Triangle, an intriguing visual pattern emerges that bears a striking resemblance to the Sierpinski Triangle.

Squares

In Pascal’s Triangle, the numbers along the third diagonal represent square numbers. Specifically, a square number is obtained by adding any two consecutive numbers within the third diagonal of the triangle.