Deriving Nth Power of (a+b): Geometrical Methods

[let_me_think]

can u tell me all possible ways of deriving nth power of (a+b) other than -- multiplying (a+b) again and again;binomial theorem and pascal triangle. CAN U TELL ME A FEW MORE METHODS? I'M PARTICULARLY INTERESTED IN GEOMETRICAL METHODS (someone told me there's one using PYTHAGORAS THEOREM). u may think I'm asking u a crazy question , but this is my holiday homework project for maths

 

[StatusX]

The binomial theorem is:

(a+b)^n = \sum_{k=0}^n \left( \begin{array}{cc} n \\ k \end{array} \right)a^k b^{n-k}

Are you looking for different ways to prove this? Or different expressions equal to the LHS? Or ways of numerically computing the LHS for specific values of a and b?

 

[garyljc]

The binomial theorem is:

(a+b)^n = \sum_{k=0}^n \left( \begin{array}{cc} n \\ k \end{array} \right)a^k b^{n-k}

Are you looking for different ways to prove this? Or different expressions equal to the LHS? Or ways of numerically computing the LHS for specific values of a and b?

Looks GOOD to me =)

 

[Gokul43201]

I think the OP is looking for a proof of the theorem. The obvious ones are the inductive proof and some kind of combinatoric proof (I can think of one, and I imagine others of this kind are essentially the same). I can't, however, imagine a proof based on Pythgoras.

 

[HallsofIvy]

No, I don't think so. It seems clear that the OP is looking for different methods of finding (x+ y)ⁿ, or at least the coefficients, not just a proof of the binomial theorem. Unfortunately, I can't think of any!