Evaluate the Summation sum from k=1 to 3 of k
The question asks for the calculation of a finite mathematical summation. Specifically, it requests the sum of an arithmetic sequence where the variable "k" takes on integer values starting at 1 and going up to 3. You are supposed to add together the values of "k" for each integer within the given range, effectively evaluating the expression 1 + 2 + 3.
$\sum_{k = 1}^{3} k$
Answer
Solution:
Step 1:
Write out the terms of the series when $k$ takes on each value from 1 to 3.
$1 + 2 + 3$
Step 2:
Combine the terms to find the sum.
Step 2.1:
Combine the first two numbers.
$1 + 2 = 3$, resulting in $3 + 3$
Step 2.2:
Add the result from the previous step to the last number.
$3 + 3 = 6$, giving the final sum $6$
Knowledge Notes:
The problem at hand involves evaluating a finite arithmetic series. An arithmetic series is the sum of the terms of an arithmetic sequence, a sequence of numbers in which the difference of any two successive members is a constant. In this case, the series is very simple, consisting of the first three positive integers.
To solve the problem, we follow these steps:
Expansion of the Series: We write out each term of the series explicitly. This is a straightforward process when dealing with a small number of terms, as in this problem.
Simplification: We simplify the series by performing the addition of its terms. This is done in a step-by-step fashion to keep the process clear and organized.
Step 2.1: We start by adding the first two terms together. This is a simple arithmetic operation.
Step 2.2: We then add the result from Step 2.1 to the remaining term to get the final sum of the series.
In general, the sum of an arithmetic series can be found using the formula:
$$ S_n = \frac{n}{2}(a_1 + a_n) $$
where $S_n$ is the sum of the first $n$ terms of the series, $a_1$ is the first term, and $a_n$ is the nth term. However, for this problem, since the series is very short, it is more efficient to simply add the numbers directly.
This problem is a simple illustration of how to work with series and perform basic arithmetic operations. It serves as a foundation for understanding more complex series and sequences in mathematics.
