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.