The given sequence is:

$2 \cdot 2 + 1 + 10 \cdot 2 + 1 + 21 \cdot 2 + 1 + \cdots$

We can observe a pattern where the terms increase by an arithmetic sequence inside the multiplication by 2, plus 1:

• The first term is $(2 \cdot 2 + 1) = 5$
• The second term is $(10 \cdot 2 + 1) = 21$
• The third term is $(21 \cdot 2 + 1) = 43$
• The fourth term would be $(37 \cdot 2 + 1) = 75$, and so on.

The numbers inside the multiplication (2, 10, 21, 37, ...) form an arithmetic sequence where the difference between consecutive numbers is increasing by 9:

• 10 - 2 = 8
• 21 - 10 = 11
• 37 - 21 = 16, and so on.

However, it looks complex and doesn't match traditional sequences easily. Could you confirm the pattern or provide more clarity? Would you like me to break it down further or simplify it with assumptions?

Here are some related questions for further exploration:

1. How would the sequence behave if the differences were constant?
2. Can this sequence be expressed using an explicit formula for each term?
3. How does the total number of terms affect the sum?
4. What if the numbers inside multiplication increased at a slower rate?
5. Would altering the number outside the multiplication affect the overall sum significantly?

Tip: Identify and simplify patterns in sequences to generalize and solve efficiently.