Given the recursive sequence:

- \( a_1 = 10 \)
- \( a_{n+1} = 3a_n + 2^{n+1} \) for \( n \geq 1 \)

Let's calculate the first few terms to better understand the sequence:

1. **Calculate \( a_2 \):**
   \[
   a_2 = 3a_1 + 2^2 = 3(10) + 4 = 30 + 4 = 34
   \]

2. **Calculate \( a_3 \):**
   \[
   a_3 = 3a_2 + 2^3 = 3(34) + 8 = 102 + 8 = 110
   \]

3. **Calculate \( a_4 \):**
   \[
   a_4 = 3a_3 + 2^4 = 3(110) + 16 = 330 + 16 = 346
   \]

The pattern of this sequence involves multiplying the previous term by 3 and adding powers of 2. The recursive relationship leads to an exponential growth of the terms.

Would you like a detailed solution for finding a general formula for \( a_n \), or is there any specific part you'd like to explore further?

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Here are 5 related questions for further study:

1. How would you find the explicit formula for \( a_n \) in this recursive sequence?
2. What happens if we change the starting value \( a_1 \)? How does it affect the sequence?
3. How can we approximate the growth of this sequence as \( n \) becomes very large?
4. Can this recursive relation be modeled using a homogeneous recurrence relation?
5. How do powers of 2 influence the growth rate

Explore the recursive sequence defined by a1 = 10 and an+1 = 3an + 2^(n+1). This problem demonstrates key concepts in recursion, sequence growth, and exponential progression. Learn to calculate the first few terms and analyze the behavior of the sequence as n increases. Suitable for students in Grades 10-12.

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