To answer the question, we need to evaluate the two given summations and then find the difference between them.
### Step 1: Evaluate [tex]\(\sum_{n=0}^4(1+3n)\)[/tex]
We need to find the sum of the expression [tex]\(1 + 3n\)[/tex] as [tex]\(n\)[/tex] goes from 0 to 4:
- For [tex]\(n = 0\)[/tex]: [tex]\(1 + 3(0) = 1\)[/tex]
- For [tex]\(n = 1\)[/tex]: [tex]\(1 + 3(1) = 4\)[/tex]
- For [tex]\(n = 2\)[/tex]: [tex]\(1 + 3(2) = 7\)[/tex]
- For [tex]\(n = 3\)[/tex]: [tex]\(1 + 3(3) = 10\)[/tex]
- For [tex]\(n = 4\)[/tex]: [tex]\(1 + 3(4) = 13\)[/tex]
Now, add these values together:
[tex]\[ 1 + 4 + 7 + 10 + 13 = 35 \][/tex]
So, [tex]\(\sum_{n=0}^4(1+3n) = 35\)[/tex].
### Step 2: Evaluate [tex]\(\sum_{i=2}^6(3i-5)\)[/tex]
We need to find the sum of the expression [tex]\(3i - 5\)[/tex] as [tex]\(i\)[/tex] goes from 2 to 6:
- For [tex]\(i = 2\)[/tex]: [tex]\(3(2) - 5 = 6 - 5 = 1\)[/tex]
- For [tex]\(i = 3\)[/tex]: [tex]\(3(3) - 5 = 9 - 5 = 4\)[/tex]
- For [tex]\(i = 4\)[/tex]: [tex]\(3(4) - 5 = 12 - 5 = 7\)[/tex]
- For [tex]\(i = 5\)[/tex]: [tex]\(3(5) - 5 = 15 - 5 = 10\)[/tex]
- For [tex]\(i = 6\)[/tex]: [tex]\(3(6) - 5 = 18 - 5 = 13\)[/tex]
Now, add these values together:
[tex]\[ 1 + 4 + 7 + 10 + 13 = 35 \][/tex]
So, [tex]\(\sum_{i=2}^6(3i-5) = 35\)[/tex].
### Step 3: Find the difference between the two sums
We have:
[tex]\[ \sum_{n=0}^4(1+3n) = 35 \][/tex]
[tex]\[ \sum_{i=2}^6(3i-5) = 35 \][/tex]
The difference between the two sums is:
[tex]\[ 35 - 35 = 0 \][/tex]
Therefore, the difference between [tex]\(\sum_{n=0}^4(1+3n)\)[/tex] and [tex]\(\sum_{i=2}^6(3i-5)\)[/tex] is [tex]\(0\)[/tex].
Among the given options, the correct answer is:
[tex]\[ 0 \][/tex]
