Final answer:
To calculate the sum of the first 38 terms of the given arithmetic sequence, we identified the first term and the common difference, determined the 38th term, and then applied the arithmetic sum formula to arrive at the sum of 68324.
Step-by-step explanation:
To find the sum of the first 38 terms of an arithmetic sequence, we can use the formula for the sum of the first n terms of an arithmetic sequence, which is Sn = n/2 × (a1 + an), where a1 is the first term, an is the nth term, and n is the number of terms.
Looking at the sequence given: -89, 13, 115, we can see that this is an arithmetic progression where the first term a1 is -89 and the common difference d can be found by subtracting the first term from the second term: d = 13 - (-89) = 102.
To find the 38th term a38, we use the formula for the nth term of an arithmetic sequence, which is an = a1 + (n - 1) × d. So, a38 = -89 + (38 - 1) × 102 = -89 + 37 × 102 = -89 + 3774 = 3685.
Now we can find the sum of the first 38 terms using the sum formula: S38 = 38/2 × (-89 + 3685) = 19 × 3596 = 68324.
Therefore, the sum of the first 38 terms of the sequence is 68324.