Let's analyze each part:

Part (a): Predicting the Next Numbers in the Sequence

The sequence is: $2, 7, -3, 2, -8, -3, -13, -8, -18, \ldots$

Let's examine the differences between terms to see if there's a pattern:

1. $7 - 2 = +5$
2. $-3 - 7 = -10$
3. $2 - (-3) = +5$
4. $-8 - 2 = -10$
5. $-3 - (-8) = +5$
6. $-13 - (-3) = -10$
7. $-8 - (-13) = +5$
8. $-18 - (-8) = -10$

The differences alternate between $+5$ and $-10$. Following this pattern, the next two terms should be:

1. $-18 + 5 = -13$
2. $-13 - 10 = -23$

So, the next two terms are -13 and -23.

Part (b): Predicting the Next Symbol

The pattern of symbols shown is: $M, \heartsuit, 8, M, 4, \heartsuit, \ldots$

Observing the repetition, it seems like the symbols alternate between three main symbols in the order $M, \heartsuit, 8$, followed by $M, 4, \heartsuit$. This suggests a repeating cycle of six symbols.

Following this cycle, the next two symbols after $M, \heartsuit, 8, M, 4, \heartsuit$ would be:

1. 8
2. M

Summary

The answers are:

• (a): -13, -23
• (b): 8, M

Would you like more details or have any questions?

Here are five related questions to deepen understanding:

1. What is the difference between inductive and deductive reasoning?
2. Can you find another sequence that alternates patterns?
3. How would you identify a repeating pattern in a more complex sequence?
4. What are common types of patterns found in symbol sequences?
5. Why might alternating differences be useful in real-world applications?

Tip: When identifying patterns, look for changes between terms first.