Expert: Steve Holleran - 11/20/2007

Question
This is my assignment:  Create a proof by mathematical induction that demonstrates that the sum of the first n even numbers is equal to n(n + 1).  I am taking college classes on line and can not seem to understand.  I start out with 2+4+6....because the proof asks for even numbers.  The next step is 2+4+6....n+(n+1) = n(n+1)  am I right so far?


Answer
Hi Shelly,

Okay, for induction, we first show its true for n=1:

2 = 1(1+1) True

Assume true for n= k:

2 + 4 + 6 + ... + a(k) = k(k+1)

Now show true for k+1:

2 + 4 + 6 + ... + k+1= (k+1)((k+1) + 1)= (k+1)(k+2)

Okay then ,

2 + 4 + 6 + ... + k + a(k+1)= (2+4+6+...+k) + z(k+1)

                         =      [k(k+1)] + a(k+1)

but a(k+1) = 2(k+1), since the first even number, 2 = 2(1), the second, 4 = 2(2), the third, 6=2(3), etc.,so this equals

  k(k+1) + 2(k+1) = (k+1)(k+2)

Hope you can follow this okay

Steve
                         
 k(k+1) + (