Let's evaluate the given summation [tex]\(\sum_{n=0}^4 (2n - 1)\)[/tex] step-by-step.
First, we need to understand what a summation is. The symbol [tex]\(\sum\)[/tex] (sigma) denotes summation, and it tells us to add up a series of terms. For [tex]\(\sum_{n=0}^4 (2n - 1)\)[/tex], we are summing the expression [tex]\(2n - 1\)[/tex] as [tex]\(n\)[/tex] ranges from 0 to 4.
We'll start by calculating each term in the series:
1. For [tex]\(n = 0\)[/tex]:
[tex]\[
2 \cdot 0 - 1 = 0 - 1 = -1
\][/tex]
2. For [tex]\(n = 1\)[/tex]:
[tex]\[
2 \cdot 1 - 1 = 2 - 1 = 1
\][/tex]
3. For [tex]\(n = 2\)[/tex]:
[tex]\[
2 \cdot 2 - 1 = 4 - 1 = 3
\][/tex]
4. For [tex]\(n = 3\)[/tex]:
[tex]\[
2 \cdot 3 - 1 = 6 - 1 = 5
\][/tex]
5. For [tex]\(n = 4\)[/tex]:
[tex]\[
2 \cdot 4 - 1 = 8 - 1 = 7
\][/tex]
Now, we add up these values:
[tex]\[
-1 + 1 + 3 + 5 + 7
\][/tex]
Next, to ensure the steps are clear, we add them sequentially:
[tex]\[
-1 + 1 = 0
\][/tex]
[tex]\[
0 + 3 = 3
\][/tex]
[tex]\[
3 + 5 = 8
\][/tex]
[tex]\[
8 + 7 = 15
\][/tex]
Therefore, the value of [tex]\(\sum_{n=0}^4 (2n - 1)\)[/tex] is:
[tex]\[
15
\][/tex]
