Binomial Theorem


   
 
Summary
  • A sentence is called a statement if it can be adjudged as true or false. Every statement is a sentence, but a sentence may or may not be a statement.
 
  • A statement involving natural number n is generally denoted by P(n).
 
  • Principle of mathematical induction states that if P(n) is a statement involving natural number n and
 
           - P(1) is true, i.e., the statement is true for n=1.
 

          - Truth of P(k) implies the truth of P(k+1) i.e., the statement is true for n = k+1 assuming it to be true for n = k, then the statement P(n) is true for all natural numbers.

 
 
 
  • A binomial is an algebraic expression of two terms which are connected by the operations '+' or '-'.
 
  • The binomial theorem for natural numbers states that
 
         
 
           Here, a and b may be any numbers.
 
  • Pascal's triangle
 
         The coefficients of various terms in (a+b)n for different values of n follows the pattern given below:
 
                  
 
  • General term
 
         For 0 < r < n, Tr+1 in the expression of (a + b)n is given by Tr+1 = nCran-rbr.
 
  • (r + 1)th term at the end in the expansion of (a+b)n is same as the (r + 1)th term at the beginning in (b+a)n.
 
  • Middle terms
 
        - If n is an even natural number, then there is only one middle term in the expansion of (a + b)n and is given by
 
        - If n is an odd natural number, then there are two middle terms in the expansion of (a + b)n and are given by
 
  • i) The sum of all binomial coefficients in the expansion of (1+x)n is 2n.
 
       
 
        ii) The sum of all even binomial coefficients in the expression of (1+x)n is 2n-1.
 
       
 
        iii) The sum of all odd binomial coefficients in the expansion of
 
       
 
        The last terms in (ii) and (iii) depends upon the fact whether n is even or odd.
 
  • The binomial theorem for fractional index states that
 
          
 
  • General term
 
        For r 0, Tr+1 in the expansion of (1+x)n, |x|<1,n Q is given by
 
       
 
  • If x be so small that its squares and higher powers may be neglected, then (1+x)n= 1 + nx (approximately).
 
  • If x be so small that its cube and higher powers may be neglected, then
 
  
     
   
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