To solve the problem of finding the sum of the first 24 terms of an arithmetic sequence with the given terms: -2, -5, -8, -11, ..., we need to use the formulas associated with arithmetic sequences.
### Step-by-Step Solution:
Step 1: Identify the first term ([tex]\(a\)[/tex]) and the common difference ([tex]\(d\)[/tex])
- The first term ([tex]\(a\)[/tex]) is the initial term of the sequence, which is -2.
- The common difference ([tex]\(d\)[/tex]) can be found by subtracting the first term from the second term. In this case, [tex]\(d = -5 - (-2) = -3\)[/tex].
Step 2: Determine the number of terms ([tex]\(n\)[/tex])
- We are given that we need the sum of the first 24 terms, thus [tex]\(n = 24\)[/tex].
Step 3: Find the 24th term of the sequence
- The formula to find the nth term ([tex]\(a_n\)[/tex]) of an arithmetic sequence is given by:
[tex]\[
a_n = a + (n-1) \cdot d
\][/tex]
Plugging in the values:
[tex]\[
a_{24} = -2 + (24-1) \cdot (-3)
\][/tex]
[tex]\[
= -2 + 23 \cdot (-3)
\][/tex]
[tex]\[
= -2 + (-69)
\][/tex]
[tex]\[
= -71
\][/tex]
So, the 24th term of the sequence is -71.
Step 4: Calculate the sum of the first 24 terms
- The formula for the sum ([tex]\(S_n\)[/tex]) of the first [tex]\(n\)[/tex] terms of an arithmetic sequence is:
[tex]\[
S_n = \frac{n}{2} \cdot (a + a_n)
\][/tex]
Plugging in the values:
[tex]\[
S_{24} = \frac{24}{2} \cdot (-2 + (-71))
\][/tex]
[tex]\[
= 12 \cdot (-73)
\][/tex]
[tex]\[
= -876
\][/tex]
Thus, the sum of the first 24 terms of the sequence is -876.
### Conclusion:
- The 24th term of the sequence is -71.
- The sum of the first 24 terms is -876.0.
