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I have a set of sucessive elements $\{a_1, a_2, a_3, a_4\}$ from an arithmetic progression and I need to sum all the possible product-combinations of two elements, i.e. $$ a_1\cdot a_2+a_1\cdot a_3+a_1\cdot a_4 +a_2\cdot a_3+a_2\cdot a_4+a_3\cdot a_4 $$ Is there a standard notation for this?

Thanks.

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  • $\begingroup$ There are a couple of standard ways, but sometimes the best thing is to just use words, e.g., "the sum of all products of two elements." $\endgroup$
    – user153918
    Feb 18, 2015 at 16:15

3 Answers 3

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$$\sum_{1\le i\lt j\le4} a_ia_j$$

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Yes, the sigma notation for this series is:

$$\begin{align} \sum_{j=1}^3\;\sum_{k=j+1}^4 a_j a_k \;&=\; a_1 a_2 + a_1 a_3 + a_1 a_4 + a_2 a_3 +a_2 a_4 + a_3 a_4 \\ & = \sum_{1\leq j < k \leq 4} a_j a_k \end{align}$$

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  • $\begingroup$ The first notation is best because it's concise and much more clearly conveys there are two iterators being looped through. The third notation requires much more careful parsing. $\endgroup$
    – user153918
    Feb 18, 2015 at 16:17
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I don't know of a specific notation, but your sum is

$$\frac12\left(\left(\sum a_i\right)^2-\sum a_i^2\right)$$

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