To find the sum of the first 50 terms of the given sequence [tex]\(a_n = 3n + 2\)[/tex], follow these steps:
1. Identify the general formula of the sequence:
The sequence is given by:
[tex]\[
a_n = 3n + 2
\][/tex]
2. Determine the first term and the 50th term:
- The first term ([tex]\(a_1\)[/tex]) is calculated by substituting [tex]\(n = 1\)[/tex]:
[tex]\[
a_1 = 3(1) + 2 = 5
\][/tex]
- The 50th term ([tex]\(a_{50}\)[/tex]) is calculated by substituting [tex]\(n = 50\)[/tex]:
[tex]\[
a_{50} = 3(50) + 2 = 152
\][/tex]
3. Use the formula for the sum of the first [tex]\(n\)[/tex] terms of an arithmetic series:
The formula for the sum of the first [tex]\(n\)[/tex] terms ([tex]\(S_n\)[/tex]) of an arithmetic series is:
[tex]\[
S_n = \frac{n}{2} (a_1 + a_n)
\][/tex]
Here, [tex]\(n = 50\)[/tex], [tex]\(a_1 = 5\)[/tex], and [tex]\(a_{50} = 152\)[/tex].
4. Substitute the values into the formula and calculate the sum:
[tex]\[
S_{50} = \frac{50}{2} (5 + 152)
\][/tex]
Simplify the expression inside the parentheses first:
[tex]\[
5 + 152 = 157
\][/tex]
Then, continue simplifying the formula:
[tex]\[
S_{50} = 25 \times 157
\][/tex]
Multiplying the values:
[tex]\[
S_{50} = 3925
\][/tex]
So, the sum of the first 50 terms of the sequence [tex]\(a_n = 3n + 2\)[/tex] is
[tex]\[
3925
\][/tex]
