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

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In summary, there are three main methods for deriving the nth power of (a+b): multiplying (a+b) repeatedly, using the binomial theorem, and using Pascal's triangle. However, the OP is looking for additional methods, particularly geometric ones such as using the Pythagorean theorem. They are interested in finding different ways to prove the binomial theorem or alternative expressions that are equal to the left-hand side. Unfortunately, no other methods are known at this time.
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let_me_think
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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
 
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  • #2
The binomial theorem is:

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

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?
 
  • #3
StatusX said:
The binomial theorem is:

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

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 =) :-p
 
  • #4
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.
 
  • #5
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!
 

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

What is the formula for deriving the Nth power of (a+b)?

The formula for deriving the Nth power of (a+b) using geometrical methods is (a+b)^N = Sum(k=0 to N) (n choose k)*(a^k)*(b^(N-k)), where (n choose k) is the binomial coefficient.

How do you use geometrical methods to derive the Nth power of (a+b)?

Geometrical methods involve using visual representations, such as diagrams or shapes, to understand mathematical concepts. To derive the Nth power of (a+b), one can use the Pascal's triangle or the binomial theorem to visualize the coefficients and exponents involved in the formula.

What is the significance of deriving the Nth power of (a+b) using geometrical methods?

Using geometrical methods to derive the Nth power of (a+b) can provide a deeper understanding of the concept and its relationship with other mathematical concepts. It can also be helpful in solving more complex problems and applications in various fields of science and engineering.

Are there any limitations to using geometrical methods for deriving the Nth power of (a+b)?

While geometrical methods can provide a visual representation of the formula, it may not always be the most efficient or practical method for deriving the Nth power of (a+b). In some cases, algebraic methods may be more suitable and accurate.

How can the Nth power of (a+b) be applied in real-world situations?

The Nth power of (a+b) has various applications in fields such as physics, engineering, and finance. For example, it can be used to calculate the sum of a series of numbers, determine the probability of certain outcomes, or model growth and decay in natural phenomena.

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