To solve the sum [tex]\(\sum_{n=2}^6 (3n + 2)\)[/tex], we need to evaluate the expression inside the summation for each value of [tex]\(n\)[/tex] from 2 to 6 and then add up the results.
Let's evaluate the expression for each [tex]\(n\)[/tex] step by step:
1. For [tex]\(n = 2\)[/tex]:
[tex]\[
3(2) + 2 = 6 + 2 = 8
\][/tex]
2. For [tex]\(n = 3\)[/tex]:
[tex]\[
3(3) + 2 = 9 + 2 = 11
\][/tex]
3. For [tex]\(n = 4\)[/tex]:
[tex]\[
3(4) + 2 = 12 + 2 = 14
\][/tex]
4. For [tex]\(n = 5\)[/tex]:
[tex]\[
3(5) + 2 = 15 + 2 = 17
\][/tex]
5. For [tex]\(n = 6\)[/tex]:
[tex]\[
3(6) + 2 = 18 + 2 = 20
\][/tex]
Now, we add all these results together:
[tex]\[
8 + 11 + 14 + 17 + 20
\][/tex]
Performing the addition step-by-step:
[tex]\[
8 + 11 = 19
\][/tex]
[tex]\[
19 + 14 = 33
\][/tex]
[tex]\[
33 + 17 = 50
\][/tex]
[tex]\[
50 + 20 = 70
\][/tex]
Thus, the sum [tex]\(\sum_{n=2}^6 (3n + 2)\)[/tex] is:
[tex]\[
\boxed{70}
\][/tex]
