https://file.aiquickdraw.com/mathbot/file/1725510345327j4up2gwq.jpg

The image shows a sequence of points plotted on a graph, and the task involves the following steps:

1. Add two more points to the sequence.
2. Identify the type of sequence.
3. Write a recursive and explicit equation for the sequence.

### Step 1: Add Two More Points

We observe the current points:
- (0, 20)
- (1, 10)
- (2, 5)

From these points, it appears that the sequence is decreasing by a factor of 2 for each step. This suggests the sequence might be geometric with a common ratio of 1/2.

Thus, the next two points can be calculated:
- \( x = 3 \), \( y = 5 \times \frac{1}{2} = 2.5 \)
- \( x = 4 \), \( y = 2.5 \times \frac{1}{2} = 1.25 \)

The two additional points are:
- (3, 2.5)
- (4, 1.25)

### Step 2: Type of Sequence

This is a **geometric sequence**, where each term is multiplied by the common ratio \( r = \frac{1}{2} \).

### Step 3: Recursive and Explicit Equations

- **Recursive formula**:
  The recursive formula for a geometric sequence is:
  \[
  a_n = r \times a_{n-1}, \quad a_0 = 20
  \]
  Where \( r = \frac{1}{2} \), so the recursive formula is:
  \[
  a_n = \frac{1}{2} \times a_{n-1}, \quad a_0 = 20
  \]

- **Explicit formula**:
  The explicit formula for a geometric sequence is:
  \[
  a_n = a_0 \times r^n
  \]
  Substituting the values \( a_0 = 20 \) and \( r = \frac{1}{2} \), we get:
  \[
  a_n = 20 \times \left( \frac{1}{2} \right)^n
  \]

Let me know if you have any questions or want further clarification!

---

Here are 5 relative questions to consider:
1. How do you determine the common ratio for a geometric sequence?
2. What happens to the sequence if the common ratio is greater than 1?
3. How would the recursive formula change if the sequence was arithmetic instead of geometric?
4. Can you find the 10th term of the sequence using the explicit formula?
5. How can you visualize a geometric sequence on a graph more easily?

**Tip**: When identifying a geometric sequence, always check the ratio between consecutive terms—this is key to finding the common ratio.

Add two more points on the graph that fit with the sequence, identify the type of sequence, and write a recursive and explicit equation for each sequence.

This problem involves identifying a geometric sequence from a graph and calculating its recursive and explicit equations. The points decrease by a factor of 1/2, and the formulas used help find future terms in the sequence.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation