What is a formula for the nth term of the given sequence?
minus, 7, comma, minus, 12, comma, minus, 17, point, point, .
−7,−12,−17...
Answer
Multiple Choice Answers
a, start subscript, n, end subscript, equals, minus, 7, left parenthesis, minus, 5, right parenthesis, start superscript, n, minus, 1, end superscripta 
n
​	
 =−7(−5) 
n−1
 
a, start subscript, n, end subscript, equals, minus, 12, minus, 5, left parenthesis, n, minus, 1, right parenthesisa 
n
​	
 =−12−5(n−1)
a, start subscript, n, end subscript, equals, minus, 7, plus, 5, left parenthesis, n, minus, 1, right parenthesisa 
n
​	
 =−7+5(n−1)
a, start subscript, n, end subscript, equals, 3, minus, 5, left parenthesis, n, plus, 1, right parenthesisa 
n
​	
 =3−5(n+1)

The given sequence is: -7, -12, -17, ...

Let's find the formula for the nth term by analyzing the pattern.

1. **Common difference**: 
   The difference between consecutive terms is:
   \[
   -12 - (-7) = -5
   \]
   \[
   -17 - (-12) = -5
   \]
   The common difference is -5, so the sequence is an **arithmetic sequence** with a difference of -5.

2. **Formula for the nth term of an arithmetic sequence**:
   The general formula for the nth term of an arithmetic sequence is:
   \[
   a_n = a_1 + (n - 1) \cdot d
   \]
   where:
   - \( a_n \) is the nth term,
   - \( a_1 \) is the first term,
   - \( d \) is the common difference,
   - \( n \) is the term number.

3. **Substitute the values**:
   - First term \( a_1 = -7 \),
   - Common difference \( d = -5 \).

   So the formula becomes:
   \[
   a_n = -7 + (n - 1) \cdot (-5)
   \]

4. **Simplify the expression**:
   \[
   a_n = -7 - 5(n - 1)
   \]
   \[
   a_n = -7 - 5n + 5
   \]
   \[
   a_n = -2 - 5n
   \]

Thus, the formula for the nth term is:
\[
a_n = -7 + 5(n - 1)
\]

**Correct answer:**
\[
a_n = -7 + 5(n - 1)
\]

This corresponds to option **C**.

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Do you want further details or have any questions? Here are 5 related questions:

1. How can we derive the common difference from any arithmetic sequence?
2. What is the general form of an arithmetic sequence, and how does it apply to this example?
3. Can the common difference in an arithmetic sequence be positive or negative, and how does that affect the pattern?
4. How do you verify if a sequence is arithmetic?
5. How does the formula change if the common difference is different?

**Tip:** When finding the nth term of a sequence, always check for a common difference to determine if it's arithmetic.

What is a formula for the nth term of the given sequence? −7,−12,−17...

Discover how to find the nth term of the arithmetic sequence −7,−12,−17... using the arithmetic sequence formula. This solution explains the pattern, common difference, and provides a step-by-step guide to the formula. Suitable for students in Grades 6-8.

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