Exhaustion Methods Explained: Sum of Areas of Outer & Inner Rectangles

[andrewkg]

Well if anyone has read Apostol's vol 1. Can anyone explain in the beginingthe sum of the areas of the outer rectangles is (b^3/n^3)(k^2) or
(b^3/n^3)(1^2+2^2+3^2…+n^2)

And the area of the inner rectangles and (b^3/n^3)(1^2+2^2+3^2…+(n-1)^2)

What I would like to know is why does the value of k for the inner area sum goes to n-1
Thank you

 

[andrewkg]

Oh yeah. My idea was that the interior rectangles cannot have a value directly at b so the value of k must be under the of n so n doesn't cancek out. Although this woukd not occur anyways because the value of n on the bottom is n^3 and the value of k on tome is only squared. Honestly though I don't have a good idea. I tryed it graphically on my own paper I got lost with what I was doing.

 

[HallsofIvy]

You should know that
1^2+ 2^2+ 3^2+ \cdot\cdot\cdot+ n^2= \frac{n(n+1)(2n+1)}{6}

As to what happens to the "k", I doubt if anyone can answer because you have not said what "k" represents.

 

[andrewkg]

Nvm I got it. Thanks though