Answer :
To find the next three terms of the arithmetic sequence given [tex]\(-18, -13, -8, -3,\)[/tex], let's follow a step-by-step process:
### Step-by-Step Solution
1. Identify the first term ([tex]\(a\)[/tex]) and the common difference ([tex]\(d\)[/tex])
- The first term, denoted [tex]\(a\)[/tex], is [tex]\(-18\)[/tex].
- The common difference [tex]\(d\)[/tex] is calculated by subtracting the first term from the second term:
[tex]\( d = -13 - (-18) \)[/tex]
[tex]\[ d = -13 + 18 = 5 \][/tex]
2. Use the common difference to find the next terms
- The next term in the sequence can be found by adding the common difference to the last known term.
- The last known term provided in the sequence is [tex]\(-3\)[/tex].
3. Calculate the subsequent terms
[tex]\[ \text{Next term (5th term)} = -3 + 5 = 2 \][/tex]
[tex]\[ \text{Next term (6th term)} = 2 + 5 = 7 \][/tex]
[tex]\[ \text{Next term (7th term)} = 7 + 5 = 12 \][/tex]
Therefore, the next three terms of the sequence following [tex]\(-3\)[/tex] are [tex]\(2\)[/tex], [tex]\(7\)[/tex], and [tex]\(12\)[/tex].
### Compare with the given options:
- a. -8, -13, -18: This option does not follow the sequence as calculated.
- b. 3, 8, 13: This option does not match our calculated terms.
- c. 2, 7, 12: This option correctly matches our calculated terms.
- d. 0, 5, 10: This option does not match our calculated terms.
### Conclusion
The correct answer is option c. 2, 7, 12.
### Step-by-Step Solution
1. Identify the first term ([tex]\(a\)[/tex]) and the common difference ([tex]\(d\)[/tex])
- The first term, denoted [tex]\(a\)[/tex], is [tex]\(-18\)[/tex].
- The common difference [tex]\(d\)[/tex] is calculated by subtracting the first term from the second term:
[tex]\( d = -13 - (-18) \)[/tex]
[tex]\[ d = -13 + 18 = 5 \][/tex]
2. Use the common difference to find the next terms
- The next term in the sequence can be found by adding the common difference to the last known term.
- The last known term provided in the sequence is [tex]\(-3\)[/tex].
3. Calculate the subsequent terms
[tex]\[ \text{Next term (5th term)} = -3 + 5 = 2 \][/tex]
[tex]\[ \text{Next term (6th term)} = 2 + 5 = 7 \][/tex]
[tex]\[ \text{Next term (7th term)} = 7 + 5 = 12 \][/tex]
Therefore, the next three terms of the sequence following [tex]\(-3\)[/tex] are [tex]\(2\)[/tex], [tex]\(7\)[/tex], and [tex]\(12\)[/tex].
### Compare with the given options:
- a. -8, -13, -18: This option does not follow the sequence as calculated.
- b. 3, 8, 13: This option does not match our calculated terms.
- c. 2, 7, 12: This option correctly matches our calculated terms.
- d. 0, 5, 10: This option does not match our calculated terms.
### Conclusion
The correct answer is option c. 2, 7, 12.
