factorials, square roots, and infinite series

a few math questions

in the binomial expansion theroem equation, basically:

Code:

`(a+b)!`

--------

a! * b!

where a+b is the exponent to multiply out a (x+y) binormial, and b+1 is the b'th element in that line in pascal's triangle

how is that equation derived?

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is there a way to get a square root of a non-square number without a bunch of trial and error?

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in an infinite geometric series, the equation for the sum is:

where r is between 1 and -1, and a is the first element. how is this equation derived? i can only get this far:

Code:

`sum = a + a*r + a*r^2 + a*r^3 ...`

= a(1 + r + r^2 + r^3 ...)