Binomial Theorem


   
 
Principle of Mathematical Induction
If P(n) is a statement (n N); such that:
 
  • P(1) is true and
 
  • truth of P(k) implies the truth of P(k+1), then by the principle of
 
          mathematical induction (P.M.I.), the statement P(n) is true for n N.
 
          Working rules for using P.M.I.
 
Step 1: Show that the result holds for n = 1.
 
Step 2: Assume the validity of the result for n equal to some arbitrary but fixed natural number, say, k.
 
Step 3: Show that the result holds for n = k+1.
 
 
Step 4: Conclude that the result holds for all natural numbers.
 
Using Principle of Mathematical Induction
 
For any natural number n, prove that
 
 
Proof:
 
We shall use PMI to prove that
 
 
Step 1: Let n = 1
 
 
 
(1) is true for n = 1.
 
Step 2: Let (1) be true for n = k
 
…(2)
 
Step 3: Let n = k + 1
 
LHS of (1)
 
 
 
 
 
 
 
 
R.H.S of (1)
 
\ (1) is true for n = k + 1.
 
By P.M.I., we have
 
 
  
     
   
Get FREE Live Tutoring
Get FREE Live Tutoring
(No credit card required)