To evaluate the summation [tex]\(\sum_{n=1}^{14} (3n + 2)\)[/tex], follow these steps:
1. Identify the general term of the sequence: The general term given in the summation is [tex]\(3n + 2\)[/tex].
2. List the terms in the series: To better understand the summation, list out the terms for [tex]\(n\)[/tex] ranging from 1 to 14.
- When [tex]\(n = 1\)[/tex], the term is [tex]\(3(1) + 2 = 3 + 2 = 5\)[/tex].
- When [tex]\(n = 2\)[/tex], the term is [tex]\(3(2) + 2 = 6 + 2 = 8\)[/tex].
- When [tex]\(n = 3\)[/tex], the term is [tex]\(3(3) + 2 = 9 + 2 = 11\)[/tex].
- When [tex]\(n = 4\)[/tex], the term is [tex]\(3(4) + 2 = 12 + 2 = 14\)[/tex].
- When [tex]\(n = 5\)[/tex], the term is [tex]\(3(5) + 2 = 15 + 2 = 17\)[/tex].
- When [tex]\(n = 6\)[/tex], the term is [tex]\(3(6) + 2 = 18 + 2 = 20\)[/tex].
- When [tex]\(n = 7\)[/tex], the term is [tex]\(3(7) + 2 = 21 + 2 = 23\)[/tex].
- When [tex]\(n = 8\)[/tex], the term is [tex]\(3(8) + 2 = 24 + 2 = 26\)[/tex].
- When [tex]\(n = 9\)[/tex], the term is [tex]\(3(9) + 2 = 27 + 2 = 29\)[/tex].
- When [tex]\(n = 10\)[/tex], the term is [tex]\(3(10) + 2 = 30 + 2 = 32\)[/tex].
- When [tex]\(n = 11\)[/tex], the term is [tex]\(3(11) + 2 = 33 + 2 = 35\)[/tex].
- When [tex]\(n = 12\)[/tex], the term is [tex]\(3(12) + 2 = 36 + 2 = 38\)[/tex].
- When [tex]\(n = 13\)[/tex], the term is [tex]\(3(13) + 2 = 39 + 2 = 41\)[/tex].
- When [tex]\(n = 14\)[/tex], the term is [tex]\(3(14) + 2 = 42 + 2 = 44\)[/tex].
3. Sum the terms: Now, add up all the terms in the series.
[tex]\(5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32 + 35 + 38 + 41 + 44\)[/tex]
4. Analyze the sum: To obtain the result more systematically, you can notice that the terms form an arithmetic sequence. However, directly adding the numbers will give the summation result:
[tex]\[
5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32 + 35 + 38 + 41 + 44 = 343
\][/tex]
Thus, the answer is [tex]\(343\)[/tex]. So, the correct answer is:
[tex]\[
\boxed{343}
\][/tex]
