# To write summation decreasing index

• MHB
In summary, to write a summation with decreasing index, you can use any of the following notations: $\sum^{1}_{k=n}x_k$, $\sum_{k=n,...,1}x_k$, $\left|\begin{array}{}s \leftarrow 0 \\\text{for }k\leftarrow n \text{ downto } 1 \\\quad s \leftarrow s + x_k \\s\end{array}\right.$, or $\sum_{k=1}^n x_{n-k+1}$. These notations are all equivalent and represent the sum of elements from index n down to 1.
Gold Member
MHB
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

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:
$$\left|\begin{array}{} s \leftarrow 0 \\ \text{for }k\leftarrow n \text{ downto } 1 \\ \quad s \leftarrow s + x_k \\ s \end{array}\right.$$

You could also write:

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

## 1. What is a summation with decreasing index?

A summation with decreasing index is a mathematical expression that represents the sum of a series of terms, where the index of the terms decreases with each subsequent term. This is usually written in the form of Σk=n to 1 a_k, where k is the index and n is the starting value.

## 2. Why is it important to specify a decreasing index in a summation?

Specifying a decreasing index in a summation is important because it allows us to clearly define the limits of the summation and ensure that all terms are included. Without a specified index, the summation may not be well-defined and could lead to incorrect results.

## 3. How do you write a summation with decreasing index?

A summation with decreasing index is typically written in the following form: Σk=n to 1 a_k. This means that the sum starts at n and goes down to 1, with k being the index and a_k representing the terms in the summation.

## 4. What is the purpose of using a decreasing index in a summation?

The purpose of using a decreasing index in a summation is to simplify and organize the mathematical expression. By specifying the starting value and the direction of the index, we can easily determine the limits of the summation and ensure that all terms are included.

## 5. Can a summation have a decreasing index in both the upper and lower limits?

Yes, a summation can have a decreasing index in both the upper and lower limits. This is written in the form of Σk=m to n a_k, where k is the index and m and n are the upper and lower limits, respectively. The summation will start at m and go down to n, with each subsequent term having a smaller index value.

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