Certainly! Let's find the general formula for the given arithmetic sequence: [tex]\(-4, -7, -10, -13, \ldots\)[/tex].
### Understanding Arithmetic Sequences
An arithmetic sequence is one where each term after the first is found by adding a constant difference ([tex]\(d\)[/tex]) to the previous term. The general formula for the [tex]\(n^{th}\)[/tex] term ([tex]\(T_n\)[/tex]) of an arithmetic sequence is typically written as:
[tex]\[ T_n = a + (n-1)d \][/tex]
Where:
- [tex]\(a\)[/tex] is the first term of the sequence.
- [tex]\(d\)[/tex] is the common difference.
- [tex]\(n\)[/tex] is the position of the term in the sequence.
### Step-by-Step Solution
#### Step 1: Identify the first term ([tex]\(a\)[/tex])
The first term of the sequence is given as:
[tex]\[ a = -4 \][/tex]
#### Step 2: Determine the common difference ([tex]\(d\)[/tex])
The common difference ([tex]\(d\)[/tex]) can be calculated by subtracting any term from the term that follows it. Let's find [tex]\(d\)[/tex]:
- Second term ([tex]\(-7\)[/tex]) minus first term ([tex]\(-4\)[/tex]):
[tex]\[ -7 - (-4) = -7 + 4 = -3 \][/tex]
- Third term ([tex]\(-10\)[/tex]) minus second term ([tex]\(-7\)[/tex]):
[tex]\[ -10 - (-7) = -10 + 7 = -3 \][/tex]
- Fourth term ([tex]\(-13\)[/tex]) minus third term ([tex]\(-10\)[/tex]):
[tex]\[ -13 - (-10) = -13 + 10 = -3 \][/tex]
Therefore, the common difference is:
[tex]\[ d = -3 \][/tex]
#### Step 3: Rewrite in the form [tex]\(T_n = dn + c\)[/tex]
The general formula for [tex]\(T_n\)[/tex] in an arithmetic sequence is:
[tex]\[ T_n = a + (n-1)d \][/tex]
Substituting [tex]\(a = -4\)[/tex] and [tex]\(d = -3\)[/tex], we get:
[tex]\[ T_n = -4 + (n-1)(-3) \][/tex]
[tex]\[ T_n = -4 - 3(n-1) \][/tex]
[tex]\[ T_n = -4 - 3n + 3 \][/tex]
[tex]\[ T_n = -3n - 1 \][/tex]
Thus, the general form [tex]\(T_n = dn + c\)[/tex] for the given sequence is:
[tex]\[ d = -3 \][/tex]
[tex]\[ c = -1 \][/tex]
### Final Answer:
The general formula for the sequence in the form [tex]\(T_n = dn + c\)[/tex] is:
[tex]\[ T_n = -3n - 1 \][/tex]
So, [tex]\(d = -3\)[/tex] and [tex]\(c = -1\)[/tex].
