I am part of an "F3" group, which is essentially a free self-ran outside morning training for men.
This morning, I ran a workout called "12 Pains of Christmas". What we did is follow the song except that the gifts were exercises (pains). So, we do one burpee, run the track, one burpee and two "flutter kicks", run the track, one burpee, two flutter kicks, and three of another exercise, then run the track, and so on until 12. Of course, every time we stopped to exercise, I would sing the song to announce all the exercises.
For fun, someone counted the reps and I was wondering if there is a way to have a formula.
The burpees are done $12\times 1$ times, the "flutter kicks" are done $11\times 2$ times, and so on. So the total reps reads: $$ S_{12}=1*12+2*11+3*8+...+12*1=364. $$
I do not want to do this only for 12 days of Christmas but for any number of days or number of sets.
Is there a formula for the following summation? That would be the case for $k+1$ days (pains), so you can do this workout for any number of days. $$ S_k=\sum_{i=1}^{k} i\,(k+1-i),\;\;\;k\geq 1. $$ The "12 pains of Christmas" would correspond to $S_{12}$.
Or, perhaps more simply, a formula for $p-1$ pains, which looks like $$ {{S}}_p'=\sum_{i=1}^{p} i\,(p-i),\;\;\;p\geq 1, $$ and we would want to ${{S}}_{13}'$. Notice in the last summation that the last term is zero, but I kept it to make the summation as simple as I could.
Also, more generally, you could start with a set of "$n$ exercises", like if you would get $n$ gifts on the first day of Christmas instead of just one. Then the summation would be $$ {{S}}_{n,p}''=\sum_{i=0}^{p-1} (i+n)\,(p-i),\;\;\;p\geq 1,n\geq 1. $$ The twelve pains as originally described correspond to ${{S}}_{1,12}''$, while starting with 3 gifts and going over 15 days correspond to ${{S}}_{3,15}''$.
Also, after the workout, people told me we should count the exercises done in "cadence", that is when I count "1-2-3" for each, and then people say "1", then I count again "1-2-3" for each, and then people say "1". So, for push-ups for example, it means that for each rep done in cadence, I really do 2 pushups. In other words, for exercises done in cadence, you do double the count. So, let say the set
$$
C=\{m_j\}_{j=1}^{c},\;\;1\leq m_j<p\text{ for all }1\leq j\leq c,
$$
contains the exercise numbers (between $1$ and $p$) done in cadence. So we need to count those exercises double.
So, we have now
$$
{{S}}_{n,p,C}'''=\sum_{i=0}^{p-1} (i+n)\,(p-i)+\sum_{j=1}^{c} (m_j-1+n)\,(p-m_j+1),\;\;\;p\geq 1,\;n\geq 1.
$$
In my case, we did the exercises 3,4,7,8, 10, and 11 in cadence, so I am looking at
${{S}}_{1,12,\{3,4,7,8,10,11\}}'''$. That is, I started with one gift, went up to 12, and did exercises 3,4,7,8,10, and 11 in cadence. Is there a way to have a nice compact formula for ${{S}}_{n,p,C}'''$?
While it's true that my first question is similar to that one, that latter question was closed because of lack of context. Mine has context. On top of that, that deleted question has no chosen answer. I was also directed to a second question for my second more general problem. Also, a closed not answered problem. I kept it there for readability (so I do not do the most general problem on the first step). The guys who worked out with me asked me to include the cadences. So, I added that in the last generalization.
