To solve this, let's first determine the sequence step by step.

Given the rule: multiply the previous number by $-2$ and add $3$.

Let $a_4$ (the fourth term) be $-7$. To find the previous term, $a_3$:

$a_4 = -2 \times a_3 + 3$

Substitute $a_4 = -7$:

$-7 = -2 \times a_3 + 3$

Solve for $a_3$:

$-7 - 3 = -2 \times a_3$

$-10 = -2 \times a_3$

$a_3 = \frac{-10}{-2} = 5$

Now, find $a_2$:

$a_3 = -2 \times a_2 + 3$

Substitute $a_3 = 5$:

$5 = -2 \times a_2 + 3$

$5 - 3 = -2 \times a_2$

$2 = -2 \times a_2$

$a_2 = \frac{2}{-2} = -1$

Finally, find $a_1$:

$a_2 = -2 \times a_1 + 3$

Substitute $a_2 = -1$:

$-1 = -2 \times a_1 + 3$

$-1 - 3 = -2 \times a_1$

$-4 = -2 \times a_1$

$a_1 = \frac{-4}{-2} = 2$

Now that we have the first four terms: $2$, $-1$, $5$, and $-7$, we can find the next three terms by applying the rule.

1. $a_5 = -2 \times (-7) + 3 = 14 + 3 = 17$
2. $a_6 = -2 \times 17 + 3 = -34 + 3 = -31$
3. $a_7 = -2 \times (-31) + 3 = 62 + 3 = 65$

So, the next three terms in the sequence are:

$\boxed{17}, \boxed{-31}, \boxed{65}$

Would you like any further details or explanations?

Related Questions:

1. How can sequences be classified by their rules?
2. What is the general formula for finding the nth term in a geometric sequence?
3. How do arithmetic and geometric sequences differ?
4. Can sequences have negative terms, and how do they affect the rule?
5. What are some common applications of sequences in real-world scenarios?

Tip: When dealing with sequences, always check the rule carefully, as small changes in the rule can significantly affect the terms of the sequence.