Expert: Steve Holleran - 6/5/2007

Question
prove, by induction, that 2 + 7 + 12+...+(5n-3)= 1/2N(5N-1)

Answer
Hello Mario,

Okay, so we have to show two things:

I.  The proposition is true for n = 1

II. Assuming its true for n = k, we can show it true for
   n = k+1


For n = 1:  5(1) - 3 = 1/2 * (1) * (5*1 - 1)

                  2 = 1/2 * 1 * 4 = 2      True.

For n = k :

2 + 7 + ... + (5k-3) = 1/2 * k * (5k-1)

and for k-->k+1:

2 + 7 + ... + (5k-3)+ (5(k+1) -3)= 1/2 * (k+1) * (5(k+1)-1)

On the left, we can replace the part
2 + 7 + ... + (5k-3) with 1/2 * k * (5k-1), and get

  1/2 * k * (5k-1) + (5(k+1)-3)

=  1/2 * k * (5k-1) + (5k +2)

= 5/2 * k^2 - 1/2 *  k + 5k + 2

= 5/2 * k^2 + 9/2 * k + 2.

Now, on the right side of the k-->k+1 equation:

 1/2 * (k+1) * (5(k+1) - 1)

= 1/2 * (k+1) * (5k + 4)

= 1/2 * (5k^2 + 9k +4)

= 5/2 * k^2 + 9/2 * k + 2  = the left side, so the proposition is proven.

Hope this is what you needed.

Steve Holleran