Certainly! Let's break down how to find the sum of the first 6 terms of the given arithmetic sequence: [tex]\(-3, 1, 5, 9, ...\)[/tex]
1. Identify the first term ([tex]\(a\)[/tex]) and the common difference ([tex]\(d\)[/tex]) of the arithmetic sequence.
- The first term ([tex]\(a\)[/tex]) is [tex]\(-3\)[/tex].
- The common difference ([tex]\(d\)[/tex]) can be found by subtracting the first term from the second term: [tex]\(1 - (-3) = 4\)[/tex].
2. List the first six terms of the sequence.
- We start with the first term, which is [tex]\(-3\)[/tex].
- The second term is obtained by adding the common difference to the first term: [tex]\(-3 + 4 = 1\)[/tex].
- The third term is obtained by adding the common difference to the second term: [tex]\(1 + 4 = 5\)[/tex].
- The fourth term is obtained by adding the common difference to the third term: [tex]\(5 + 4 = 9\)[/tex].
- The fifth term is obtained by adding the common difference to the fourth term: [tex]\(9 + 4 = 13\)[/tex].
- The sixth term is obtained by adding the common difference to the fifth term: [tex]\(13 + 4 = 17\)[/tex].
Hence, the first six terms of the sequence are: [tex]\(-3, 1, 5, 9, 13, 17\)[/tex].
3. Calculate the sum of these six terms.
- Sum the terms: [tex]\(-3 + 1 + 5 + 9 + 13 + 17\)[/tex].
4. Add the numbers step-by-step:
[tex]\[
-3 + 1 = -2
\][/tex]
[tex]\[
-2 + 5 = 3
\][/tex]
[tex]\[
3 + 9 = 12
\][/tex]
[tex]\[
12 + 13 = 25
\][/tex]
[tex]\[
25 + 17 = 42
\][/tex]
5. Confirm the sum of the first six terms.
- The sum is [tex]\(42\)[/tex].
Thus, the correct answer is:
A) 42
