How to generalize the following operation in one mathematical equation

Question

I'm trying to generalize the following formulations :

$$score_1=\frac{\sum_{j=1}^{3}FN(v_1,s_1)+FN(v_1,s_2)+FN(v_1,s_3)}{3}$$ $$score_2=\frac{\sum_{j=1}^{3}FN(v_2,s_1)+FN(v_2,s_2)+FN(v_2,s_3)}{3}$$

I have tried the following equation :
lets say N=4 While 1 <= i <= N $$score_i=\frac{\sum_{j=1}^{k}FN(v_i,s_j)}{k}$$

is my approach right ?

Answer 1

Hint: It seems the expressions $score_1$ and $score_2$ do not really match what you had in mind. Nevertheless taking it verbatim we obtain

\begin{align*} score_1&=\frac{\sum_{j=1}^{3}FN(v_1,s_1)+FN(v_1,s_2)+FN(v_1,s_3)}{3}\\ &=\frac{3FN(v_1,s_1)+FN(v_1,s_2)+FN(v_1,s_3)}{3}\tag{1}\\ &=FN(v_1,s_1)+\frac{1}{3}\sum_{j=2}^3FN(v_1,s_j)\tag{2}\\ score_2&=FN(v_2,s_1)+\frac{1}{3}\sum_{j=2}^3FN(v_2,s_j)\tag{3}\\ \end{align*}

We generalise (1) and (3) by iterating the first argument $v_i (1\leq i \leq N)$ and taking $k\geq 1$ as upper index of the sum. We obtain \begin{align*} score_i&=FN(v_i,s_1)+\frac{1}{k}\sum_{j=2}^kFN(v_i,s_j)\qquad\qquad 1\leq i\leq N, k\geq 1\tag{4} \end{align*}

Comment:

• In (1) the scope of the sum encloses the term $FN(v_1,s_1)$ but no other terms. Since this term does not depend on the index $j$ and is treated as a constant according to the rule $\sum_{j=1}^n a=a\sum_{j=1}^n 1 =n\cdot a$.

• In (2) we simplify the expression somewhat.

• In (3) we collect the terms besides the left-hand one in a sum.

• In (4) observe, that in case $k=1$ we have $\sum_{j=2}^\color{blue}{1}FN(v_i,s_j)=0$, since the upper limit $1$ of the sum is less than the lower limit $2$.

Hint: To me it seems you had something different in mind, namely

\begin{align*} score_1&=\frac{FN(v_1,s_1)+FN(v_1,s_2)+FN(v_1,s_3)}{3}\\ &=\frac{1}{3}\left(FN(v_1,s_1)+FN(v_1,s_2)+FN(v_1,s_3)\right)\\ &=\frac{1}{3}\sum_{j=1}^3FN(v_1,s_j)\\ score_2&=\frac{1}{3}\sum_{j=1}^3FN(v_2,s_j)\\\\ \end{align*}

The generalisation gives now

\begin{align*} \color{blue}{score_i}&\color{blue}{=\frac{1}{k}\sum_{j=1}^kFN(v_i,s_j)\qquad\qquad 1\leq i\leq N,\ k\geq 1} \end{align*}

If for instance (4) is not that clear, a helpful, thorough exposition of how to work with sums is presented in chapter 2: Sums , section 2.1 Notation in Concrete Mathematics by R.L. Graham, D.E. Knuth and O. Patashnik.

Answer 2

The first two expressions are written without $\sum _{j=1}^3$ (because they're already written explicitly). The expression $\mathrm{score}_i$ is correctly written.

If we specify what $i$ is, we usually say "for all $i$ such that $1\leq i\leq N$" or just "for $1\leq i\leq N$". While $1\leq i \leq N$ is totally acceptable in my eyes, but opinions may vary on this.

Take care to not overload your notation. For instance, at the moment I'm detecting $N$ and $N(x,y)$ or is it $FN(x,y)$?