# Thread: to write summation decreasing index

1. how to write a summation with decreasing index

say for adding from index 1 to n for $x_k$ we write $\sum^{n}_{k=1}x_k$.

how do we write the above for index to go from n to 1 down wards

2.

3. Mathematically they are identical, aren't they?
$k$ really iterates over a set of elements, which is unordered.
So there wouldn't be a separate mathematical notation for it.

Improvising, we might write:
$$\sum^{1}_{k=n}x_k$$
or:
$$\sum_{k=n,...,1}x_k$$
or:
$$\left|\begin{array}{} s \leftarrow 0 \\ \text{for }k\leftarrow n \text{ downto } 1 \\ \quad s \leftarrow s + x_k \\ s \end{array}\right.$$

4. You could also write:

$\displaystyle \sum_{k=1}^n x_{n-k+1}$