Prove by using mathematical induction that 1+2+3+...+n= [n(n+1)]/2, n is a member of N.

Question

Prove by using mathematical induction 

Answer 1

N=1:

 

  1 = 1(1+1)/2 = 1*2/2 = 1*1 = 1

 

 

N=2:

   1+2 = 3 = 2(3)/2 = 2(1+3)/2

 

So the statement is true for N=1,2

 

 

Induction Hypothesis:

  (GIVEN)   1 + 2 + 3 + ..... + k-1  + k = k(k+1)/2

 

WTS (Wish to show/prove):   1 + 2 + 3 + ..... + k-1 + k + k+1 = (k+1)(k+2)/2

 

 

Proof:

 

     1 + 2 + 3 + ..... + k-1 + k + k+1 =   k(k+1)/2 + (k+1)  <--- given by induction hypothesis

 

                                                        =  k(k+1)/2 + 2(k+1)/2  <--- common denominator

 

                                                         = k(k+1)+2(k+1)]/2   <--- numerators can be added

 

                                                         = [ (k+1)(k+2)/2 ] <--- factors out k+1 in the numerator

 

 

[End of Proof : Statement holds by induction for all positive integers n=1,2,3,.....]