Given a pyramid with square base $b\times b$ and height $h$ we start by slicing the pyramid parallel to the base with $n$ slices of thickness $h/n$. A typical slice has one face as a square $(i/n)b\times (i/n)b$ and the parallel face a square $((i-1)/n)b\times ((i-1)/n)b$, so the volume of the slice is more than $h/n\times ((i-1)/n)b\times ((i-1)/n)b$ but less than $h/n\times (i/n)b\times (i/n)b$.
Add the slices to show that
$\frac{hb^{2}}{n^{3}}(1^{2}+2^{2}+\cdots+(n-1)^{2}) Source: Numbers and Functions by R. P. Burn. Chapter: 3 Question: 25 I've got two problems: The question says a typical slice has base of length $(i/n)b$. What I got from that is $i/n$ is some fraction of b. I'm guessing $i$ is some sort of variable which changes as we pick different bases for a slice of the pyramid. Is $i/n$ derived from properties of a pyramid or is it an arbitrary choice to represent a fraction of $b$? The next thing I'm having trouble with is figuring out which slices to add. Putting aside my confusion from the last point I can see that the volume of a 'smaller' slice is $\frac{hb^{2}}{n^{3}}(i-1)^{2}$ and a 'bigger' slice is $\frac{hb^{2}}{n^{3}}(i)^{2}$. But after this point I'm a bit lost as how to proceed to get the result asked for in the question. Any help would be much appreciated.