Which explicit formula defines the given arithmetic sequence? (General form: an = a1 + d(n − 1)

Responses

A

an = 24 + 10n

a n = 24 + 10n

B

an = −34 + 10n

a n = −34 + 10n

C

an = −24 + 10n

a n = −24 + 10n

D

an = −24 −10n

To determine which of the provided formulas defines an arithmetic sequence, we'll compare each option to the general form of the arithmetic sequence formula:

\[ a_n = a_1 + d(n − 1) \]

### Step 1: Compare Each Option to the General Form

1. **Option A:** \( a_n = 24 + 10n \)

   - Rewriting in the form \( a_n = a_1 + d(n − 1) \):
   \[
   a_n = 24 + 10n = 24 + 10n - 10 + 10 = 10n + 14
   \]
   - Here, this doesn't match the general form directly, but it can be rewritten as \( a_n = 14 + 10n \), which still doesn't quite fit the exact general form \( a_n = a_1 + d(n − 1) \).

2. **Option B:** \( a_n = −34 + 10n \)

   - Rewriting:
   \[
   a_n = -34 + 10n = -34 + 10n
   \]
   - This can be written as \( a_n = -34 + 10n - 10 + 10 = -34 + 10(n - 1) \), which fits the form better.

3. **Option C:** \( a_n = −24 + 10n \)

   - Rewriting:
   \[
   a_n = -24 + 10n = -24 + 10(n - 1)
   \]
   - This fits well since \( a_n = -24 + 10n - 10 + 10 \), so it's correct.

4. **Option D:** \( a_n = −24 − 10n \)

   - Rewriting:
   \[
   a_n = -24 - 10n = -24 + (-10)(n - 1)
   \]
   - This is a descending arithmetic sequence.

### Step 2: Match the Form
- Options **B** and **C** match the general arithmetic sequence form well, but **C** is the correct one since it most closely resembles \( a_n = a_1 + d(n − 1) \) explicitly, with \( a_1 = -24 \) and \( d = 10 \).

### Correct Answer:
**C: \( a_n = −24 + 10n \)**

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**Relative Questions:**
1. How can you rewrite a given arithmetic sequence to match the general form?
2. What is the significance of the first term in the sequence?
3. How does changing the sign of the common difference affect the sequence?
4. Which option would represent a decreasing arithmetic sequence?
5. How can you determine the common difference directly from a formula?

**Tip:** When rewriting a formula, pay close attention to matching the general arithmetic sequence structure to identify the first term and common difference correctly.

Explore how to identify the explicit formula for an arithmetic sequence using the general form an = a1 + d(n - 1). We compare different options to find the correct formula, focusing on understanding the role of the first term and the common difference. Suitable for high school students.

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