TITLE: Double index summation: $\sum_{ 1 \leq i < j \leq 3 }(2i+j). $
QUESTION [10 upvotes]: I understand multiple summations, but I do not know how to do this summation. Could someone break this down for me?
For example:
 $$\sum_{ 1 \leq i < j \leq 3 }(2i+j). $$

REPLY [2 votes]: As has been stated,
this is a double sum.
If we let the
upper limit be $n$
instead of $3$,
it can be explicitly converted
into a double sum
and then evaluated like this:
$\begin{array}\\
\sum_{ 1 \leq i < j \leq n }(2i+j)
&=\sum_{1 < j \le n}\sum_{1\le i < j}(2i+j)\\
&=\sum_{j=2}^{ n}\sum_{i=1}^{j-1}(2i+j)\\
&=\sum_{j=2}^{ n}\sum_{i=1}^{j-1}(2i)+\sum_{j=2}^{n}\sum_{i=1}^{j-1}(j)\\
&=\sum_{j=2}^{ n}2\dfrac{(j-1)j}{2}+\sum_{j=2}^{n}j(j-1)\\
&=2\sum_{j=2}^{n}j(j-1)\\
\end{array}
$
and this you should be
able to evaluate.