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Rewrite the sum so that exactly the same quantity is expressed but so that the outer sum is over the index $l$ and the inner sum is over the index $k$. Lay the quantities $x_{lk}$ out in a triangular table to see what is going on.

Lay the quantities $x_{lk}$ out in a triangular table to see what is going on. $$\sum_{k=1}^q \left[\sum_{l=1}^k x_{lk}\right]$$

I am confused mainly by writing out the table it asks for.

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For series usually it is useful to expand some terms to see what is going on. For example here:

$$\sum_{k=1}^q [\sum_{l=1}^k x_{lk}] = x_{11} + (x_{12} + x_{22}) + (x_{13} + x_{23} + x_{33}) + ... $$

If you draw a triangular table or simply check the elements that are being summed from the $q\times q$ matrix of possible values $x_{ij}$, you can see that they belong to the upper right triangular matrix:

$$ \begin{bmatrix} x_{11} & x_{12}& ... & x_{1q} \\ & x_{22} & ... & x_{2q} \\ & & & x_{qq} \\ \end{bmatrix} $$

Now you have to rearrange the indices of the sums such that the same elements are summed

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    So the matrix you wrote out refers to the first sum, not the rewritten one? I'm slightly confused with the x1q, x2q, xqq – 2017-02-08
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    Exactly. The sums are developed from "inside" to "outside" Let's see: for $k=1$, we develop all terms of the inner summation: $k=1$, and since the inner summation goes from $l=1$ to $l=k=1$, the only term of this first sum is $x_{11}$. Now for the next value of $k$, $k=2$, we have the inner summation going form $l=1$ to $l=k=2$, which expands two terms: $x_{12}$ and $x_{22}$. Hoepfully you can see the pattern. This will go up unitl the limit of the outer summation, $k=q$ – 2017-02-08