Let's determine the first three terms of the sequence using the rule: multiply the previous number by $-2$ and add $3$.

Let the first term be $a_1$, the second term be $a_2$, and so on. We know:

$a_4 = -7$

To find the terms that came before $a_4 = -7$, we work backwards:

1. Find $a_3$:
   $a_4 = -2 \times a_3 + 3$ Substitute $a_4 = -7$: $-7 = -2 \times a_3 + 3$ Solve for $a_3$: $-10 = -2 \times a_3 \implies a_3 = 5$

2. Find $a_2$:
   $a_3 = -2 \times a_2 + 3$ Substitute $a_3 = 5$: $5 = -2 \times a_2 + 3$ Solve for $a_2$: $2 = -2 \times a_2 \implies a_2 = -1$

3. Find $a_1$:
   $a_2 = -2 \times a_1 + 3$ Substitute $a_2 = -1$: $-1 = -2 \times a_1 + 3$ Solve for $a_1$: $-4 = -2 \times a_1 \implies a_1 = 2$

So, the first three terms in the sequence are:

1. $a_1 = 2$
2. $a_2 = -1$
3. $a_3 = 5$

The three terms that come before $-7$ are 2, -1, and 5.

Would you like more details or have any questions? Here are some related questions:

1. What is the fifth term in this sequence?
2. How do you find the general formula for the $n$th term in such sequences?
3. How does changing the multiplier from $-2$ to another number affect the sequence?
4. What if the rule was "divide by 2 and subtract 3" instead?
5. Can you find the sum of the first 10 terms in this sequence?

Tip: Always verify your final answers by plugging them back into the original rule to ensure consistency.