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