Certainly! Let's solve this step-by-step.
### Step 1: Identify the given values
- The first term [tex]\(a_1\)[/tex] of the arithmetic sequence is 13.
- The common difference [tex]\(d\)[/tex] is 3.
- We need to find the sum of the first 7 terms, [tex]\(s_7\)[/tex].
### Step 2: Compute the 7th term ([tex]\(a_7\)[/tex]) of the arithmetic sequence
The formula to find the [tex]\(n\)[/tex]th term of an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n - 1)d \][/tex]
For [tex]\(n = 7\)[/tex], we have:
[tex]\[ a_7 = a_1 + (7 - 1)d \][/tex]
[tex]\[ a_7 = 13 + 6 \cdot 3 \][/tex]
[tex]\[ a_7 = 13 + 18 \][/tex]
[tex]\[ a_7 = 31 \][/tex]
So the 7th term, [tex]\(a_7\)[/tex], is 31.
### Step 3: Compute the sum of the first 7 terms of the sequence
The formula to find the sum [tex]\(S_n\)[/tex] of the first [tex]\(n\)[/tex] terms of an arithmetic sequence is:
[tex]\[ S_n = \frac{n}{2} \cdot (a_1 + a_n) \][/tex]
For [tex]\(n = 7\)[/tex], we have:
[tex]\[ S_7 = \frac{7}{2} \cdot (a_1 + a_7) \][/tex]
[tex]\[ S_7 = \frac{7}{2} \cdot (13 + 31) \][/tex]
[tex]\[ S_7 = \frac{7}{2} \cdot 44 \][/tex]
[tex]\[ S_7 = 7 \cdot 22 \][/tex]
[tex]\[ S_7 = 154 \][/tex]
### Step 4: Select the correct answer
From the calculations, the sum of the first 7 terms [tex]\(S_7\)[/tex] is 154. Hence, the correct answer is:
[tex]\[ \boxed{154} \][/tex]
