Find A Closed Formula For 1 2 0 3 T Pascal’s Triangle and Square Numbers

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Pascal’s Triangle and Square Numbers

Below are the first few rows of Pascal’s triangle:

1 1

1 2 1

13 3 1

14 6 4 1

15 10 10 5 1

16 15 21 15 6 1

The bold numbers are the third diagonal when Pascal’s triangle is drawn at the center. These are triangular numbers, which are formed by the sum of consecutive whole numbers (eg 15 = 1 + 2 + 3 + 4 + 5), and from these we can form square numbers. We have to add consecutive numbers from these and we get square numbers. To get the first square number, we add 0 to the front of the list:

0, 1, 3, 6, 10, 15, 21…

0 + 1 = 1 = 1^2

1 + 3 = 4 = 2^2

3+ 6 = 9 = 3^2

6 + 10 = 16 = 4^2

10 + 15 = 25 = 5^2

15 + 21 = 36 = 6^2

Incidentally, you can also get square numbers by taking the differences of two places on the fourth diagonal of Pascal’s triangle. The fourth diagonal is 1, 4, 10, 20 35… , and the differences you get are 1-0 = 1, 4-0 = 4, 10-1 = 9, 20-4 = 16, 35-10 = . 25 and so on.

To understand why you get square numbers by adding consecutive triangle numbers together, you can use different methods. First, if you know that the formula for the nth triangular number is (n^2 + n)/2, then the previous triangular number n is less than this, because it is the same sum of numbers but (n-1) and what you add Not n as the last number. If we add these two numbers together, we get

(n^2 + n)/2 + (n^2 + n)/2 – n

= (1/2)n^2 + n/2 + (1/2)n^2 + n/2 – n

= n^2 + n – n

= n^2

If that method is not to your liking, we can show this result graphically. Triangle numbers get their name from the fact that you can make them by adding square numbers from the number of dots that make up different shapes of triangles and the number of dots that make up squares of different shapes. So we just have to make a square from two triangle points. If you do it using coins or counters, or paper, and make right-angled triangles, you can make a square out of two triangles, but one counter must be smaller on each side of it. Well, it wasn’t a rigorous method to prove it, but it was a lot easier than doing a lot of algebra, right?

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