To find the sum of the series [tex]\(\sum_{k=3}^8(2k + 2)\)[/tex], follow these steps:
1. Identify the Range of [tex]\(k\)[/tex]:
The range for [tex]\(k\)[/tex] is from 3 to 8. Thus, [tex]\(k\)[/tex] takes the values: 3, 4, 5, 6, 7, and 8.
2. Determine the Expression:
The given series is [tex]\(\sum_{k=3}^8(2k + 2)\)[/tex]. We need to evaluate [tex]\(2k + 2\)[/tex] for each [tex]\(k\)[/tex] within the specified range.
3. Evaluate the Expression for Each [tex]\(k\)[/tex]:
- For [tex]\(k = 3\)[/tex]:
[tex]\[ 2 \cdot 3 + 2 = 6 + 2 = 8 \][/tex]
- For [tex]\(k = 4\)[/tex]:
[tex]\[ 2 \cdot 4 + 2 = 8 + 2 = 10 \][/tex]
- For [tex]\(k = 5\)[/tex]:
[tex]\[ 2 \cdot 5 + 2 = 10 + 2 = 12 \][/tex]
- For [tex]\(k = 6\)[/tex]:
[tex]\[ 2 \cdot 6 + 2 = 12 + 2 = 14 \][/tex]
- For [tex]\(k = 7\)[/tex]:
[tex]\[ 2 \cdot 7 + 2 = 14 + 2 = 16 \][/tex]
- For [tex]\(k = 8\)[/tex]:
[tex]\[ 2 \cdot 8 + 2 = 16 + 2 = 18 \][/tex]
4. List the Terms:
The terms to be summed are:
[tex]\[ 8, 10, 12, 14, 16, 18 \][/tex]
5. Sum the Terms:
Adding these terms together:
[tex]\[ 8 + 10 + 12 + 14 + 16 + 18 \][/tex]
Let's break it down:
[tex]\[ 8 + 10 = 18 \][/tex]
[tex]\[ 18 + 12 = 30 \][/tex]
[tex]\[ 30 + 14 = 44 \][/tex]
[tex]\[ 44 + 16 = 60 \][/tex]
[tex]\[ 60 + 18 = 78 \][/tex]
Therefore, the sum of the series is [tex]\(78\)[/tex].
6. Final Answer:
The correct option is:
[tex]\[ 8 + 10 + 12 + 14 + 16 + 18 ; 78 \][/tex]
