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BINOMIAL THEOREM FOR POSITIVE INTEGRAL INDEX

If x and a are real , then

i.e

• General term for is
  
  
• In the binomial expansion of , term from end is term from beginning .
• If n is odd then no .of terms in . Have equal no. of terms =
• If n is odd then has terms has terms
• Middle term in a binomial expansion
  If n is even
  Then middle term of is term
• If n is odd then Middle terms are and terms
• STANDARD NOTATIONS
  
  
• Examples based on Integer part and fraction part

• where
  
  I ,n Then (I+f) ( I –f) = 1
  
  Steps to solve these type problems:
  
  1) write the given expression =I +F
  
  Where I is Integer ,F is fractional part
  
  2) Define G by replacing ‘+’ sign by ‘-‘.Note G always lies between 0 and 1
  
  3) Either add or subtract G from expression in Step -I so that R.H.S is an integer.
  
  4) G+F = 1 i.e. if G is added G = 1-F
  
  i.e if G is subtracted then G-F = 0 = G =F
  
  5) Obtain the value of desired expression given.
  
  

Greatest term

Let and and is integral part then

1. If is not a integer , Then is numerically greatest term in the binomial expansion of

2. If is an integer then and are numerically greatest terms in expansion

Largest binomial coefficient :

The largest among is (are ) :

1. If n is event integer then is largest.