what the recusrsive formula for the sequance of 15 26 48 92 and 180

To find the recursive formula for the sequence \(15, 26, 48, 92, 180\), we need to determine the pattern or relationship between the terms.

Let's denote the sequence by \(a_n\), where:
- \(a_1 = 15\)
- \(a_2 = 26\)
- \(a_3 = 48\)
- \(a_4 = 92\)
- \(a_5 = 180\)

We need to find a recursive relation of the form \(a_{n+1} = f(a_n)\).

### Step-by-Step Approach

1. **Calculate Differences:**
   \[
   a_2 - a_1 = 26 - 15 = 11
   \]
   \[
   a_3 - a_2 = 48 - 26 = 22
   \]
   \[
   a_4 - a_3 = 92 - 48 = 44
   \]
   \[
   a_5 - a_4 = 180 - 92 = 88
   \]

2. **Calculate Ratios of Differences:**
   \[
   \frac{22}{11} = 2
   \]
   \[
   \frac{44}{22} = 2
   \]
   \[
   \frac{88}{44} = 2
   \]

The ratios of the differences are constant and equal to 2, suggesting the differences double each time.

3. **Formulate the Recursive Formula:**
   Since the differences double each time, we can represent the difference in a recursive manner. Let \(d_n\) be the difference between consecutive terms \(a_n\) and \(a_{n+1}\). Then:
   \[
   d_1 = 11
   \]
   \[
   d_{n+1} = 2d_n
   \]

Given that \(a_{n+1} = a_n + d_n\), and \(d_1 = 11\), we can write:
   \[
   a_{n+1} = a_n + 11 \cdot 2^{n-1}
   \]

### Verification
- For \(a_2\):
  \[
  a_2 = a_1 + 11 \cdot 2^{1-1} = 15 + 11 \cdot 1 = 26
  \]
- For \(a_3\):
  \[
  a_3 = a_2 + 11 \cdot 2^{2-1} = 26 + 11 \cdot 2 = 48
  \]
- For \(a_4\):
  \[
  a_4 = a_3 + 11 \cdot 2^{3-1} = 48 + 11 \cdot 4 = 92
  \]
- For \(a_5\):
  \[
  a_5 = a_4 + 11 \cdot 2^{4-1} = 92 + 11 \cdot 8 = 180
  \]

This confirms that the recursive formula works.

### Recursive Formula
The recursive formula for the sequence is:
\[ a_{n+1} = a_n + 11 \cdot 2^{n-1} \]

Would you like details on this or have any questions?

Here are some related questions to expand on this topic:

1. What is the closed form of the given recursive sequence?
2. How do you find the \(n\)th term directly using the closed form?
3. How do you prove that the given recursive formula is correct?
4. What are some other sequences that have a similar recursive pattern?
5. How can you determine the recursive formula for a given sequence?
6. What is the difference between a linear and a geometric recursive sequence?
7. How do you use recursive sequences in real-world applications?
8. Can you find a recursive formula for a sequence where the ratios of differences are not constant?

**Tip:** Understanding how to derive a recursive formula is crucial for sequences where a direct formula is not immediately apparent. Practice with different types of sequences to get more comfortable with this process.

Discover the recursive formula for the sequence 15, 26, 48, 92, 180 with a step-by-step explanation. Learn how to derive and verify the recursive relation using differences and mathematical induction. Suitable for advanced high school students.

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