Let's analyze and expand the given summation series:
[tex]\[
\sum_{n=0}^5 6n
\][/tex]
This notation tells us to sum the values of [tex]\(6n\)[/tex] as [tex]\(n\)[/tex] ranges from 0 to 5. Here are the step-by-step calculations:
1. For [tex]\(n = 0\)[/tex]:
[tex]\[
6 \times 0 = 0
\][/tex]
2. For [tex]\(n = 1\)[/tex]:
[tex]\[
6 \times 1 = 6
\][/tex]
3. For [tex]\(n = 2\)[/tex]:
[tex]\[
6 \times 2 = 12
\][/tex]
4. For [tex]\(n = 3\)[/tex]:
[tex]\[
6 \times 3 = 18
\][/tex]
5. For [tex]\(n = 4\)[/tex]:
[tex]\[
6 \times 4 = 24
\][/tex]
6. For [tex]\(n = 5\)[/tex]:
[tex]\[
6 \times 5 = 30
\][/tex]
Now, let's list these terms to form the expanded series and verify the sum:
[tex]\[
0 + 6 + 12 + 18 + 24 + 30
\][/tex]
The expanded form of the given series is:
[tex]\[
[0, 6, 12, 18, 24, 30]
\][/tex]
Additionally, summing these terms:
[tex]\[
0 + 6 + 12 + 18 + 24 + 30 = 90
\][/tex]
So, the expanded form is [tex]\([0, 6, 12, 18, 24, 30]\)[/tex] and the total sum of the series is [tex]\(90\)[/tex].
