Evaluate the Summation sum from x=1 to 3 of x

Solution:

Step 1:

Write out the terms of the series by substituting $x$ with each integer from $1$ to $3$.

$1 + 2 + 3$

Step 2:

Combine the terms.

Step 2.1:

Sum the first two numbers: $1 + 2$ equals $3$.

$3 + 3$

Step 2.2:

Add the result from Step 2.1 to the last number: $3 + 3$ equals $6$.

$6$

Knowledge Notes:

The problem at hand is a simple arithmetic summation, which involves finding the sum of all values in a sequence from the starting point to the endpoint. The sequence in this case is a series of integers starting from $1$ to $3$.

The process of evaluating this summation involves two main knowledge points:

1. Arithmetic Series: An arithmetic series is the sum of the terms of an arithmetic sequence, a sequence of numbers in which each term after the first is obtained by adding a constant difference to the previous term. In this case, the series is simple and does not require the use of the arithmetic series sum formula, as the number of terms is small.

2. Summation Notation: The summation notation $\sum$ is used to represent the sum of a sequence of numbers. The notation includes an expression for the terms to be added, a variable that takes on successive values in a set, and limits of summation that specify the first and last values that the variable will take. In this problem, the summation is from $x=1$ to $x=3$ of $x$, which means we add up the values of $x$ as it takes on each integer value from $1$ to $3$.

The solution to the problem is straightforward and involves basic addition. The steps are clearly outlined, first by expanding the series to show all the terms that need to be added, and then by simplifying through addition. The process is methodical, ensuring that each term is accounted for and that the final sum is reached systematically.