The given sequence is:
1, 5, 13, 29, 61, 125, ...

To find the recursive formula, we need to observe the relationship between consecutive terms.

Let's denote the terms of the sequence as $a_n$, where $a_1 = 1$, $a_2 = 5$, $a_3 = 13$, and so on.

Step 1: Calculate the differences between consecutive terms:

$a_2 - a_1 = 5 - 1 = 4$ $a_3 - a_2 = 13 - 5 = 8$ $a_4 - a_3 = 29 - 13 = 16$ $a_5 - a_4 = 61 - 29 = 32$ $a_6 - a_5 = 125 - 61 = 64$

This shows that the differences between consecutive terms are: 4, 8, 16, 32, 64,...

Step 2: Observe the pattern in the differences:

The differences form a geometric sequence where each term is twice the previous one. That is: $\Delta a_n = 2^{n+1}$

Step 3: Establish the recursive relation:

Since the difference between consecutive terms follows a pattern, the recursive formula for the sequence is:

$a_{n+1} = a_n + 2^{n+1}$ with the initial condition: $a_1 = 1$

This is the recursive formula for the given sequence.

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:

1. How would you derive the explicit formula for this sequence?
2. What is the sum of the first $n$ terms of this sequence?
3. How can the behavior of the geometric difference be generalized for other sequences?
4. Can you find a different recursive relation for other similar sequences?
5. How does this recursive relation help in finding terms for large $n$?

Tip: Recursive formulas are often easier to derive and work with when there's a clear pattern in differences between consecutive terms.