To find the recursive formula for the given sequence $2, 4, 10, 24, 58$, we need to analyze how the sequence progresses and find a pattern that we can express recursively.

Step 1: First differences

Start by calculating the first differences between consecutive terms:

$4 - 2 = 2$ $10 - 4 = 6$ $24 - 10 = 14$ $58 - 24 = 34$

So, the first differences are: $2, 6, 14, 34$.

Step 2: Second differences

Next, calculate the second differences between consecutive first differences:

$6 - 2 = 4$ $14 - 6 = 8$ $34 - 14 = 20$

The second differences are: $4, 8, 20$.

Step 3: Third differences

Now, calculate the third differences between consecutive second differences:

$8 - 4 = 4$ $20 - 8 = 12$

The third differences are: $4, 12$.

Step 4: Fourth differences

Finally, calculate the fourth differences:

$12 - 4 = 8$

Since the fourth differences are constant, the sequence has a recursive relation that involves a second-order difference.

Step 5: Finding the recursive relation

From the differences, we can guess a recursive relation of the form:

$a_n = a_{n-1} + d_n$ where $d_n$ is some function of $n$. Based on the pattern of the differences, we can propose a recursive formula.

Conclusion

The exact recursive formula for this sequence is complex, but we can express the recursive relation as:

$a_n = 2a_{n-1} + a_{n-2} + \text{constant term adjustments}$

Would you like a more detailed step-by-step exploration of this recursive process, or do you have any questions on a particular part of the solution?

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Follow-up Questions:

1. How can we express the second-order recursive relation more clearly?
2. What is the significance of constant differences in sequences?
3. How would we use this recursive formula to generate future terms in the sequence?
4. What happens when the sequence involves higher-order differences (i.e., third or higher)?
5. How do we generalize this method to other sequences?

Tip: When analyzing sequences, always start by calculating the differences between terms to identify any patterns that can lead to a recursive relationship.