Pi From Pascal's Triangle

This program (made with python 3) derives the number pi (3.14159... etc.) from Pascal's triangle.

Each number in the triangle is calculated by adding together the two numbers above it.

Pi is derived from Pascal's triangle by taking the reciprocal of the third number of each row (excluding the first two rows), changing the sign of each fraction by alternating between two positive and two negative signs, starting with positive (+ + - - + + - - etc.), adding the fractions together, and them adding two to the sum.

For example, the first few reciprocals would be:

Applying the + + - - part, we have

Adding these together yeilds approximately 1.14207181707.

Adding 2 to this yeilds 3.14207181707, which is equal to pi to 2 decimal places.

The more numbers are used, the more accurate the result is. For example, when 10,000 numbers are used in the program, the result is 3.141592633593785, which is more accurate than the previous example.