**Instructions:** For the following sequence, state the common difference, identify which is the explicit form and which is the recursive form of the rule, and find the term listed.

**Sequence:** 10,−190,−390,−590,…10,−190,−390,−590,…

**Common Difference:** Answer 1 Question 8

an=210−200nan=210−200n

Answer 2 Question 8 ExplicitRecursive

an=an−1−200an=an−1−200

a1=10


### Sequence: 10, −190, −390, −590, ...

#### Step 1: Find the common difference
The common difference (denoted as \( d \)) in an arithmetic sequence is found by subtracting consecutive terms.

\[
d = -190 - 10 = -200
\]

So, the common difference is \( -200 \).

#### Step 2: Identify the explicit and recursive forms

The explicit form gives a direct formula to calculate any term in the sequence without needing the previous terms. The recursive form defines each term based on the previous term(s).

- **Explicit form:**
  
  The explicit formula for this sequence is:

  \[
  a_n = 210 - 200n
  \]

  This formula allows us to calculate the \(n\)-th term directly, so this is the **explicit form**.

- **Recursive form:**

  The recursive formula is:

  \[
  a_n = a_{n-1} - 200
  \]
  \[
  a_1 = 10
  \]

  This rule defines each term based on the previous one, so this is the **recursive form**.

#### Step 3: Find the specified term

To find a specific term, you can use either the explicit or recursive formula, depending on which term you're looking for.

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Would you like to know how to use these formulas for a specific term or have further details?

Here are 5 questions to expand the information:

1. How do we use the recursive form to calculate the 5th term?
2. How do we convert between explicit and recursive forms?
3. Why is the common difference important in arithmetic sequences?
4. Can the recursive formula be used to calculate any term directly?
5. What is the geometric interpretation of an arithmetic sequence?

**Tip:** Always check your common difference carefully when working with sequences to avoid errors in calculations!

For the following sequence, state the common difference, identify which is the explicit form and which is the recursive form of the rule, and find the term listed. Sequence: 10, −190, −390, −590,…

Learn how to find the common difference and identify the explicit and recursive formulas for the arithmetic sequence: 10, -190, -390, -590. Explore key concepts like arithmetic sequences and their formulas. Suitable for students in Grades 9-11.

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