The sequence in the image is:

$1, 11, 27, 49, 77, 111, \, ?$

To find the next number, let's analyze the differences between consecutive terms:

• $11 - 1 = 10$
• $27 - 11 = 16$
• $49 - 27 = 22$
• $77 - 49 = 28$
• $111 - 77 = 34$

The differences are increasing by 6 each time:

$10, 16, 22, 28, 34$

Thus, the next difference should be:

$34 + 6 = 40$

Adding this to the last term of the sequence:

$111 + 40 = 151$

So, the next number in the sequence is 151.

Would you like more details or have any questions?

Here are some related questions to expand on this:

1. Can you find a general formula for this sequence?
2. How would you classify this type of sequence?
3. Can the differences between terms give any clues about patterns in more complex sequences?
4. How does this problem relate to arithmetic progressions?
5. Could you apply this approach to sequences where the differences follow a different rule?

Tip: When looking at sequences, always check for patterns in differences or ratios between terms—it often reveals hidden structure.