To solve the problem of finding the sum of the arithmetic series [tex]\(5 + 8 + 11 + \ldots + 122\)[/tex], let's follow these steps:
1. Identify the terms of the series:
- The first term [tex]\(a\)[/tex] is 5.
- The common difference [tex]\(d\)[/tex] is 3.
- The last term [tex]\(l\)[/tex] is 122.
2. Determine the number of terms ([tex]\(n\)[/tex]) in the series.
- The general form for the [tex]\(n\)[/tex]-th term of an arithmetic sequence is given by:
[tex]\[
a_n = a + (n-1)d
\][/tex]
- We know the last term [tex]\(a_n\)[/tex] is 122, so we need to solve for [tex]\(n\)[/tex]:
[tex]\[
122 = 5 + (n-1) \cdot 3
\][/tex]
- Subtract 5 from both sides:
[tex]\[
117 = (n-1) \cdot 3
\][/tex]
- Divide both sides by 3:
[tex]\[
39 = n-1
\][/tex]
- Add 1 to both sides:
[tex]\[
n = 40
\][/tex]
3. Calculate the sum of the series:
- The formula for the sum ([tex]\(S_n\)[/tex]) of an arithmetic series is:
[tex]\[
S_n = \frac{n}{2} \cdot (a + l)
\][/tex]
- Substitute the values we have:
[tex]\[
S_{40} = \frac{40}{2} \cdot (5 + 122)
\][/tex]
- Simplify inside the parentheses:
[tex]\[
S_{40} = 20 \cdot 127
\][/tex]
- Multiply to find the sum:
[tex]\[
S_{40} = 2540
\][/tex]
So, the sum of the series [tex]\(5 + 8 + 11 + \ldots + 122\)[/tex] is [tex]\(\boxed{2540}\)[/tex].
