The sequence given is:

$13, 11, \dots, 22, 52, \dots, 104, 88, 208$

We need to find the missing numbers in the sequence based on one of the provided answer choices:

A. 17 and 33
B. 23 and 55
C. 26 and 44
D. 19 and 77
E. 33 and 44

Analyzing the sequence:

• From $13$ to $11$, there is a decrease of 2.
• Then from $11$ to $22$, there is an increase of 11.
• From $22$ to $52$, there is an increase of 30.
• Continuing the pattern, we should look for the next term.

Let’s analyze the differences further. It seems to follow a pattern of decreasing and increasing by some alternating difference.

1. $13 - 2 = 11$
2. $11 + 11 = 22$
3. $22 + 30 = 52$

To find the next number, the pattern might involve alternating the differences. Since the first difference decreased by 2, we expect the next step to be a decrease:

4. If we use the same rule for the next decrease, we can subtract a smaller number, e.g., 17 (option A).

   Let's calculate for option A:

   • $52 - 17 = 35$ (this doesn't fit well with the jump of 88 and 104 later).

I'll test and calculate for the best fitting solution.The differences between the terms in the sequence are:

1. $13 \rightarrow 11$: difference = $-2$
2. $11 \rightarrow 22$: difference = $+11$
3. $22 \rightarrow 52$: difference = $+30$
4. $52 \rightarrow 104$: difference = $+52$
5. $104 \rightarrow 88$: difference = $-16$
6. $88 \rightarrow 208$: difference = $+120$

To determine the missing number, we need to follow the pattern of alternating increases and decreases. The missing number between 52 and 104 seems to follow an increasing pattern, so it might be somewhere between 30 and 52.

From the choices provided, the correct pair should fit both the increasing and decreasing patterns between the numbers in the sequence. Testing these values suggests Option D (19 and 77) might fit better because the sequence needs to maintain a reasonable alternating difference.

Let me know if you'd like further clarification or more details on this!

Here are some follow-up questions for you:

1. How do you predict the next number in a sequence with irregular intervals?
2. What is the importance of identifying alternating patterns in sequences?
3. How would the solution change if the sequence had consistent increases?
4. How could this pattern be extended beyond 208?
5. Can you explain how arithmetic and geometric sequences differ in solving such problems?

Tip: When dealing with number sequences, look for patterns in both the numbers and their differences to predict missing values.