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General and Middle terms of a Binomial Expansion


The binomial expansion for can be expressed as
[Keep substituting k = 0,1,2................,n and use idea of summation]

The term can be called the general term of the expansion of .

Observations:
  1. If k = 1, you get the term =[ k = 0 gives the term]. 
  2. If k = r, you get the (r + 1)term of the expansion,

 term of the expansion is called the general term of the expansion. This term has many important uses in applications of binomial theorem, as we shall see in some examples.

Example 1:
Find the term in the expansion of .

Solution:

Here


Note:
  1. When involves huge numbers do not simplify (see the above example).
  2. The general term enables us to find a particular term of the expansion without finding all the terms in the expansions.
Middle Term/s
We know that in the expansion of , there are n + 1 terms.
  1. If n is even, (n+1) is odd. The middle term is term.
    Ex: In the example which is the 5term.
  2. If n is odd, then n+1 is even. Then there are two middle terms.
    The middle terms are:
    Example: In the expansion of there are 10 terms. The middle terms are
Example:
Find the middle term /s in the expansion of


Solution:
  1. has 18 (even) terms. The middle terms are


Finding the Constant Term in the Expansion of (a + b)n


In the expansion of, if there is a term which is free from the variables (i.e. only a constant) it is called a constant term. Hence, a constant term is a term which is independent of x.

Example 1:
Find the term independent of x in the expansion of

Solution:
If there is a constant term in the expansion of , it should be

Note:
If you get r as a negative quantity or a fraction, such a term (constant) cannot exist for that particular sum.

Example 2:
Find the constant term in the expansion of

Solution:


Example 3:
Find the coefficient of

Solution:





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