The given sequence is:
$10, 10, 13, 7, 17, 5, \ ??$

Let's analyze the pattern:

1. From 10 to 10: $10 - 10 = 0$ (no change).
2. From 10 to 13: $13 - 10 = +3$.
3. From 13 to 7: $7 - 13 = -6$.
4. From 7 to 17: $17 - 7 = +10$.
5. From 17 to 5: $5 - 17 = -12$.

The pattern alternates between increasing and decreasing values:

• Increase by +3, -6, +10, -12.

We can assume the next step is an increase, following the alternating pattern. The increase appears to follow an arithmetic progression: +3, +10 (the difference between these is +7).

Thus, the next increase should be $10 + 7 = +17$.

So, from 5, we increase by 17:
$5 + 17 = 22.$

Thus, the next number in the sequence is 22.

The complete sequence:

$10, 10, 13, 7, 17, 5, 22$

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. Can you identify another sequence with alternating increases and decreases?
2. What happens if we add an extra number to this sequence following the same pattern?
3. How do arithmetic progressions work in this context?
4. Could this pattern fit into a specific type of mathematical function or rule?
5. How could you write a formula for this sequence?

Tip: Always check for alternating patterns or arithmetic progressions when analyzing sequences with varying increases and decreases.