\input zb-basic
\input zb-matheduc
\iteman{ZMATH 2016e.00849}
\itemau{Treeby, David; Tang, Jenny}
\itemti{Combinatorial derivations of familiar identities.}
\itemso{Parabola 51, No. 1, 3 p., electronic only (2015).}
\itemab
From the text: Finding two ways to enumerate the same collection of objects can often give rise to useful formulae. For instance, the sum $1+2+\dots +n$ can be interpreted as the maximum number of different handshakes between $n+1$ people. The first person may shake hands with $n$ other people. The next person may shake hands with $n-1$ other people, not counting the first person again. Continuing like this gives the above sum. Another approach is to simply realise that each of the $n+1$ guests shakes hands with $n$ other guests. However, this counts handshakes twice. Therefore, $1+2+\dots +n=\frac{n(n+1)}{2}$. This article concerns a combinatorial argument that gives rise to the familiar formula for the sum of the first $n$ squares, $1^2+2^2+\dots +n^2$. For our derivation of the formula we enumerate the same collection of objects two different ways and then equate the results.
\itemrv{~}
\itemcc{K20 I30}
\itemut{combinatorics; enumeration problems; set of consecutive integers; enumerating rising sequences; sums of squares}
\itemli{https://www.parabola.unsw.edu.au/files/articles/2010-2019/volume-51-2015/issue-1/vol51\_no1\_2.pdf}
\end