mathematical induction

We wish to show that....

1 + 2 + 3 + 4 +.....+ (n-1) + n =  n(n + 1) /2

Show it's true for n= 1

(1)(1+1)/2= 2/2  = 1

Assume it's true for k, that is    1 + 2 + 3 + 4 + ..... +( k - 1) + k  = k(k+1)/2

Prove it's true for k + 1

That is  .....1 + 2 + 3 + 4 +......+ k + (k+1)  = (k+1)(k+2)/2

We have

1 + 2 + 3 + 4 +......+ k + (k+1 ) = k(k+1)/2 + (k+1) =

1 + 2 + 3 + 4 +......+ k + ( k+1)  = k(k+1)/2 + 2(k+1)/2 =

1 + 2 + 3 + 4 +......+ k + (k+1)  =  [k(k+1) + 2(k+1)] / 2 =  {factor out (k + 1) }

1 + 2 + 3 + 4 +......+ k + (k+1)  = [(k +1)(k + 2)] / 2

And that's what we wished to prove.....!!!