Answer :
To find the explicit rule for the given arithmetic sequence [tex]\(-182, -175, -168, -161, \ldots\)[/tex], we need to determine two things: the first term and the common difference.
### Step 1: Identify the First Term
The first term of the sequence is:
[tex]\[ a_1 = -182 \][/tex]
### Step 2: Find the Common Difference
The common difference ([tex]\(d\)[/tex]) is found by subtracting the first term from the second term:
[tex]\[ d = -175 - (-182) \][/tex]
Simplifying this, we have:
[tex]\[ d = -175 + 182 = 7 \][/tex]
### Step 3: Write the Explicit Formula
The general formula for the nth term of an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
Substitute the first term [tex]\(-182\)[/tex] and the common difference [tex]\(7\)[/tex] into the formula:
[tex]\[ a_n = -182 + (n - 1) \cdot 7 \][/tex]
### Step 4: Simplify the Expression
Now simplify the expression:
1. Distribute [tex]\(7\)[/tex] through [tex]\((n - 1)\)[/tex]:
[tex]\[ a_n = -182 + 7n - 7 \][/tex]
2. Combine like terms:
[tex]\[ a_n = 7n - 189 \][/tex]
Thus, the explicit formula for the sequence is:
[tex]\[ a_n = 7n - 189 \][/tex]
This matches the option [tex]\( a_n = 7n - 189 \)[/tex].
### Step 1: Identify the First Term
The first term of the sequence is:
[tex]\[ a_1 = -182 \][/tex]
### Step 2: Find the Common Difference
The common difference ([tex]\(d\)[/tex]) is found by subtracting the first term from the second term:
[tex]\[ d = -175 - (-182) \][/tex]
Simplifying this, we have:
[tex]\[ d = -175 + 182 = 7 \][/tex]
### Step 3: Write the Explicit Formula
The general formula for the nth term of an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
Substitute the first term [tex]\(-182\)[/tex] and the common difference [tex]\(7\)[/tex] into the formula:
[tex]\[ a_n = -182 + (n - 1) \cdot 7 \][/tex]
### Step 4: Simplify the Expression
Now simplify the expression:
1. Distribute [tex]\(7\)[/tex] through [tex]\((n - 1)\)[/tex]:
[tex]\[ a_n = -182 + 7n - 7 \][/tex]
2. Combine like terms:
[tex]\[ a_n = 7n - 189 \][/tex]
Thus, the explicit formula for the sequence is:
[tex]\[ a_n = 7n - 189 \][/tex]
This matches the option [tex]\( a_n = 7n - 189 \)[/tex].
