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Example:Finding Middle Term using Binomial Expansion

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Binomial theorem



The binomial coefficients appear as the entries of Pascal's triangle

The binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power (x + y)n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term. For example,

The coefficient a in the term of xbyc is known as the binomial coefficient  or  (the two have the same value). These coefficients for varying n and b can be arranged to form Pascal's triangle. These numbers also arise in combinatorics, where  gives the number of different combinations of b elements that can be chosen from an n-element set.


Statement of the theorem

According to the theorem, it is possible to expand any power of x + y into a sum of the form

where each  is a specific positive integer known as binomial coefficient. This formula is also referred to as the Binomial Formula or the Binomial Identity. Using summation notation, it can be written as

(Our solved example in mathguru.com uses this concept)




The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.


(Our solved example in mathguru.com also uses the below concept. This is our own explanation, it is not taken from Wikipedia.)



If we have to find the middle term of an expansion it depends upon the number of terms in it. The number of terms in the expansion of (a + b) n depends on the index "n". The index "n" is either even or odd.


If n is odd

 Let n = 2k+1

The number of terms is n+1 i.e., (2k + 1) + 1 = 2k + 2.

In this case, there are two middle terms and are after k terms.